Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Percentage Accurate: 89.4% → 98.7%

Time: 8.3s

Alternatives: 16

Speedup: 0.4×

Specification

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 89.4% accurate, 1.0× speedup

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := t \cdot z - x\\ \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t\_1}, y, \frac{x}{x - t \cdot z} + x\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - -1}{x + \frac{y}{t}}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)))
   (if (<= (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0)) INFINITY)
     (/ (fma (/ z t_1) y (+ (/ x (- x (* t z))) x)) (+ x 1.0))
     (/ 1.0 (/ (- x -1.0) (+ x (/ y t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double tmp;
	if (((x + (((y * z) - x) / t_1)) / (x + 1.0)) <= ((double) INFINITY)) {
		tmp = fma((z / t_1), y, ((x / (x - (t * z))) + x)) / (x + 1.0);
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0)) <= Inf)
		tmp = Float64(fma(Float64(z / t_1), y, Float64(Float64(x / Float64(x - Float64(t * z))) + x)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 / Float64(Float64(x - -1.0) / Float64(x + Float64(y / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(z / t$95$1), $MachinePrecision] * y + N[(N[(x / N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x - -1.0), $MachinePrecision] / N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      4. lower-unsound-/.f64 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x + 1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      6. add-flip N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      8. metadata-eval 89.3%

         \[\leadsto \frac{1}{\frac{x - \color{blue}{-1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      10. add-flip N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}\right)\right)}} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\mathsf{neg}\left(\color{blue}{\left(y \cdot z - x\right)}\right)}{t \cdot z - x}}} \]

      16. sub-negate-rev N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      17. lower--.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{y \cdot z}}{t \cdot z - x}}} \]

      19. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

      20. lower-*.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

   3. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - -1}{x - \frac{x - z \cdot y}{t \cdot z - x}}}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{x + \color{blue}{\frac{y}{t}}}} \]

      2. lower-/.f64 70.3%

         \[\leadsto \frac{1}{\frac{x - -1}{x + \frac{y}{\color{blue}{t}}}} \]

   6. Applied rewrites 70.3%

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{+15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{z}{t\_1} \cdot \frac{y}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - -1}{x + \frac{y}{t}}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 4e+15)
     t_2
     (if (<= t_2 INFINITY)
       (* (/ z t_1) (/ y (- x -1.0)))
       (/ 1.0 (/ (- x -1.0) (+ x (/ y t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= 4e+15) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (z / t_1) * (y / (x - -1.0));
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= 4e+15) {
		tmp = t_2;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (z / t_1) * (y / (x - -1.0));
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= 4e+15:
		tmp = t_2
	elif t_2 <= math.inf:
		tmp = (z / t_1) * (y / (x - -1.0))
	else:
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 4e+15)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(z / t_1) * Float64(y / Float64(x - -1.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(x - -1.0) / Float64(x + Float64(y / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 4e+15)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = (z / t_1) * (y / (x - -1.0));
	else
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e+15], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[(z / t$95$1), $MachinePrecision] * N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x - -1.0), $MachinePrecision] / N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4e15

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   if 4e15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. times-frac N/A

         \[\leadsto \frac{y}{1 + x} \cdot \color{blue}{\frac{z}{t \cdot z - x}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{y}{1 + x} \cdot \frac{z}{\color{blue}{t \cdot z - x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{1 + \color{blue}{x}} \]

      9. +-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      11. lower-/.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{\color{blue}{x + 1}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      13. add-flip N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - -1} \]

      15. lower--.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{-1}} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{x - -1}} \]

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      4. lower-unsound-/.f64 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x + 1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      6. add-flip N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      8. metadata-eval 89.3%

         \[\leadsto \frac{1}{\frac{x - \color{blue}{-1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      10. add-flip N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}\right)\right)}} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\mathsf{neg}\left(\color{blue}{\left(y \cdot z - x\right)}\right)}{t \cdot z - x}}} \]

      16. sub-negate-rev N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      17. lower--.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{y \cdot z}}{t \cdot z - x}}} \]

      19. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

      20. lower-*.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

   3. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - -1}{x - \frac{x - z \cdot y}{t \cdot z - x}}}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{x + \color{blue}{\frac{y}{t}}}} \]

      2. lower-/.f64 70.3%

         \[\leadsto \frac{1}{\frac{x - -1}{x + \frac{y}{\color{blue}{t}}}} \]

   6. Applied rewrites 70.3%

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := t \cdot z - x\\ t_2 := x + \frac{y \cdot z - x}{t\_1}\\ t_3 := \frac{t\_2}{x + 1}\\ t_4 := \frac{z}{t\_1}\\ \mathbf{if}\;t\_3 \leq -1000000000000:\\ \;\;\;\;\frac{y \cdot t\_4}{x - -1}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_2}{1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4 \cdot \frac{y}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - -1}{x + \frac{y}{t}}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (+ x (/ (- (* y z) x) t_1)))
        (t_3 (/ t_2 (+ x 1.0)))
        (t_4 (/ z t_1)))
   (if (<= t_3 -1000000000000.0)
     (/ (* y t_4) (- x -1.0))
     (if (<= t_3 2e-14)
       (/ t_2 1.0)
       (if (<= t_3 2.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_3 INFINITY)
           (* t_4 (/ y (- x -1.0)))
           (/ 1.0 (/ (- x -1.0) (+ x (/ y t))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = x + (((y * z) - x) / t_1);
	double t_3 = t_2 / (x + 1.0);
	double t_4 = z / t_1;
	double tmp;
	if (t_3 <= -1000000000000.0) {
		tmp = (y * t_4) / (x - -1.0);
	} else if (t_3 <= 2e-14) {
		tmp = t_2 / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4 * (y / (x - -1.0));
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = x + (((y * z) - x) / t_1);
	double t_3 = t_2 / (x + 1.0);
	double t_4 = z / t_1;
	double tmp;
	if (t_3 <= -1000000000000.0) {
		tmp = (y * t_4) / (x - -1.0);
	} else if (t_3 <= 2e-14) {
		tmp = t_2 / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_4 * (y / (x - -1.0));
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = x + (((y * z) - x) / t_1)
	t_3 = t_2 / (x + 1.0)
	t_4 = z / t_1
	tmp = 0
	if t_3 <= -1000000000000.0:
		tmp = (y * t_4) / (x - -1.0)
	elif t_3 <= 2e-14:
		tmp = t_2 / 1.0
	elif t_3 <= 2.0:
		tmp = (x - (x / t_1)) / (x + 1.0)
	elif t_3 <= math.inf:
		tmp = t_4 * (y / (x - -1.0))
	else:
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(x + Float64(Float64(Float64(y * z) - x) / t_1))
	t_3 = Float64(t_2 / Float64(x + 1.0))
	t_4 = Float64(z / t_1)
	tmp = 0.0
	if (t_3 <= -1000000000000.0)
		tmp = Float64(Float64(y * t_4) / Float64(x - -1.0));
	elseif (t_3 <= 2e-14)
		tmp = Float64(t_2 / 1.0);
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_3 <= Inf)
		tmp = Float64(t_4 * Float64(y / Float64(x - -1.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(x - -1.0) / Float64(x + Float64(y / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = x + (((y * z) - x) / t_1);
	t_3 = t_2 / (x + 1.0);
	t_4 = z / t_1;
	tmp = 0.0;
	if (t_3 <= -1000000000000.0)
		tmp = (y * t_4) / (x - -1.0);
	elseif (t_3 <= 2e-14)
		tmp = t_2 / 1.0;
	elseif (t_3 <= 2.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	elseif (t_3 <= Inf)
		tmp = t_4 * (y / (x - -1.0));
	else
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -1000000000000.0], N[(N[(y * t$95$4), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-14], N[(t$95$2 / 1.0), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$4 * N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x - -1.0), $MachinePrecision] / N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e12

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. times-frac N/A

         \[\leadsto \frac{y}{1 + x} \cdot \color{blue}{\frac{z}{t \cdot z - x}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{x + 1} \cdot \frac{z}{t \cdot z - x} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{y}{x + 1} \cdot \frac{z}{t \cdot z - x} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{y}{x + 1} \cdot \frac{z}{\color{blue}{t \cdot z - x}} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{\color{blue}{x + 1}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{z}{t \cdot z - x} \cdot y}{\color{blue}{x} + 1} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{z}{t \cdot z - x} \cdot y}{\color{blue}{x + 1}} \]

      12. *-commutative N/A

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{\color{blue}{x} + 1} \]

      13. lower-*.f64 32.9%

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{\color{blue}{x} + 1} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{x + \color{blue}{1}} \]

      15. add-flip N/A

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{x - -1} \]

      17. lower--.f64 32.9%

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{x - \color{blue}{-1}} \]

   6. Applied rewrites 32.9%

      \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{\color{blue}{x - -1}} \]

   if -1e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-14

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{\color{blue}{1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 46.2%

      \[\leadsto \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{\color{blue}{1}} \]

   if 2e-14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{x - \frac{x}{\color{blue}{t \cdot z - x}}}{x + 1} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{x - \frac{x}{t \cdot z - \color{blue}{x}}}{x + 1} \]

      4. lower-*.f64 66.0%

         \[\leadsto \frac{x - \frac{x}{t \cdot z - x}}{x + 1} \]

   4. Applied rewrites 66.0%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. times-frac N/A

         \[\leadsto \frac{y}{1 + x} \cdot \color{blue}{\frac{z}{t \cdot z - x}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{y}{1 + x} \cdot \frac{z}{\color{blue}{t \cdot z - x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{1 + \color{blue}{x}} \]

      9. +-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      11. lower-/.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{\color{blue}{x + 1}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      13. add-flip N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - -1} \]

      15. lower--.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{-1}} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{x - -1}} \]

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      4. lower-unsound-/.f64 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x + 1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      6. add-flip N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      8. metadata-eval 89.3%

         \[\leadsto \frac{1}{\frac{x - \color{blue}{-1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      10. add-flip N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}\right)\right)}} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\mathsf{neg}\left(\color{blue}{\left(y \cdot z - x\right)}\right)}{t \cdot z - x}}} \]

      16. sub-negate-rev N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      17. lower--.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{y \cdot z}}{t \cdot z - x}}} \]

      19. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

      20. lower-*.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

   3. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - -1}{x - \frac{x - z \cdot y}{t \cdot z - x}}}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{x + \color{blue}{\frac{y}{t}}}} \]

      2. lower-/.f64 70.3%

         \[\leadsto \frac{1}{\frac{x - -1}{x + \frac{y}{\color{blue}{t}}}} \]

   6. Applied rewrites 70.3%

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ t_3 := \frac{z}{t\_1}\\ \mathbf{if}\;t\_2 \leq -1000000000000:\\ \;\;\;\;\frac{y \cdot t\_3}{x - -1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + x\right)\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3 \cdot \frac{y}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - -1}{x + \frac{y}{t}}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0)))
        (t_3 (/ z t_1)))
   (if (<= t_2 -1000000000000.0)
     (/ (* y t_3) (- x -1.0))
     (if (<= t_2 5e-21)
       (* (/ -1.0 (- -1.0 x)) (+ (/ y t) x))
       (if (<= t_2 2.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_2 INFINITY)
           (* t_3 (/ y (- x -1.0)))
           (/ 1.0 (/ (- x -1.0) (+ x (/ y t))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double t_3 = z / t_1;
	double tmp;
	if (t_2 <= -1000000000000.0) {
		tmp = (y * t_3) / (x - -1.0);
	} else if (t_2 <= 5e-21) {
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3 * (y / (x - -1.0));
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double t_3 = z / t_1;
	double tmp;
	if (t_2 <= -1000000000000.0) {
		tmp = (y * t_3) / (x - -1.0);
	} else if (t_2 <= 5e-21) {
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_3 * (y / (x - -1.0));
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	t_3 = z / t_1
	tmp = 0
	if t_2 <= -1000000000000.0:
		tmp = (y * t_3) / (x - -1.0)
	elif t_2 <= 5e-21:
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x)
	elif t_2 <= 2.0:
		tmp = (x - (x / t_1)) / (x + 1.0)
	elif t_2 <= math.inf:
		tmp = t_3 * (y / (x - -1.0))
	else:
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	t_3 = Float64(z / t_1)
	tmp = 0.0
	if (t_2 <= -1000000000000.0)
		tmp = Float64(Float64(y * t_3) / Float64(x - -1.0));
	elseif (t_2 <= 5e-21)
		tmp = Float64(Float64(-1.0 / Float64(-1.0 - x)) * Float64(Float64(y / t) + x));
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_2 <= Inf)
		tmp = Float64(t_3 * Float64(y / Float64(x - -1.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(x - -1.0) / Float64(x + Float64(y / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	t_3 = z / t_1;
	tmp = 0.0;
	if (t_2 <= -1000000000000.0)
		tmp = (y * t_3) / (x - -1.0);
	elseif (t_2 <= 5e-21)
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	elseif (t_2 <= 2.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	elseif (t_2 <= Inf)
		tmp = t_3 * (y / (x - -1.0));
	else
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1000000000000.0], N[(N[(y * t$95$3), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-21], N[(N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$3 * N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x - -1.0), $MachinePrecision] / N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e12

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. times-frac N/A

         \[\leadsto \frac{y}{1 + x} \cdot \color{blue}{\frac{z}{t \cdot z - x}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{x + 1} \cdot \frac{z}{t \cdot z - x} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{y}{x + 1} \cdot \frac{z}{t \cdot z - x} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{y}{x + 1} \cdot \frac{z}{\color{blue}{t \cdot z - x}} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{\color{blue}{x + 1}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{z}{t \cdot z - x} \cdot y}{\color{blue}{x} + 1} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{z}{t \cdot z - x} \cdot y}{\color{blue}{x + 1}} \]

      12. *-commutative N/A

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{\color{blue}{x} + 1} \]

      13. lower-*.f64 32.9%

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{\color{blue}{x} + 1} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{x + \color{blue}{1}} \]

      15. add-flip N/A

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{x - -1} \]

      17. lower--.f64 32.9%

          \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{x - \color{blue}{-1}} \]

   6. Applied rewrites 32.9%

      \[\leadsto \frac{y \cdot \frac{z}{t \cdot z - x}}{\color{blue}{x - -1}} \]

   if -1e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999997e-21

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.4%

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x + \frac{y}{t}}{\color{blue}{x + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y}{t}}{x + 1}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \left(x + \frac{y}{t}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \left(x + \frac{y}{t}\right)} \]

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \cdot \left(x + \frac{y}{t}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \cdot \left(x + \frac{y}{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \cdot \left(x + \frac{y}{t}\right) \]

      9. add-flip N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \left(x + \frac{y}{t}\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(x - \color{blue}{-1}\right)\right)} \cdot \left(x + \frac{y}{t}\right) \]

      11. sub-negate-rev N/A

          \[\leadsto \frac{-1}{\color{blue}{-1 - x}} \cdot \left(x + \frac{y}{t}\right) \]

      12. lower--.f64 70.3%

          \[\leadsto \frac{-1}{\color{blue}{-1 - x}} \cdot \left(x + \frac{y}{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(x + \color{blue}{\frac{y}{t}}\right) \]

      14. +-commutative N/A

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + \color{blue}{x}\right) \]

      15. lower-+.f64 70.3%

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + \color{blue}{x}\right) \]

   8. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + x\right)} \]

   if 4.9999999999999997e-21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{x - \frac{x}{\color{blue}{t \cdot z - x}}}{x + 1} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{x - \frac{x}{t \cdot z - \color{blue}{x}}}{x + 1} \]

      4. lower-*.f64 66.0%

         \[\leadsto \frac{x - \frac{x}{t \cdot z - x}}{x + 1} \]

   4. Applied rewrites 66.0%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. times-frac N/A

         \[\leadsto \frac{y}{1 + x} \cdot \color{blue}{\frac{z}{t \cdot z - x}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{y}{1 + x} \cdot \frac{z}{\color{blue}{t \cdot z - x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{1 + \color{blue}{x}} \]

      9. +-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      11. lower-/.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{\color{blue}{x + 1}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      13. add-flip N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - -1} \]

      15. lower--.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{-1}} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{x - -1}} \]

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      4. lower-unsound-/.f64 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x + 1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      6. add-flip N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      8. metadata-eval 89.3%

         \[\leadsto \frac{1}{\frac{x - \color{blue}{-1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      10. add-flip N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}\right)\right)}} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\mathsf{neg}\left(\color{blue}{\left(y \cdot z - x\right)}\right)}{t \cdot z - x}}} \]

      16. sub-negate-rev N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      17. lower--.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{y \cdot z}}{t \cdot z - x}}} \]

      19. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

      20. lower-*.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

   3. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - -1}{x - \frac{x - z \cdot y}{t \cdot z - x}}}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{x + \color{blue}{\frac{y}{t}}}} \]

      2. lower-/.f64 70.3%

         \[\leadsto \frac{1}{\frac{x - -1}{x + \frac{y}{\color{blue}{t}}}} \]

   6. Applied rewrites 70.3%

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1000000000000:\\ \;\;\;\;\frac{z}{\left(x - -1\right) \cdot t\_1} \cdot y\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + x\right)\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{z}{t\_1} \cdot \frac{y}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - -1}{x + \frac{y}{t}}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -1000000000000.0)
     (* (/ z (* (- x -1.0) t_1)) y)
     (if (<= t_2 5e-21)
       (* (/ -1.0 (- -1.0 x)) (+ (/ y t) x))
       (if (<= t_2 2.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_2 INFINITY)
           (* (/ z t_1) (/ y (- x -1.0)))
           (/ 1.0 (/ (- x -1.0) (+ x (/ y t))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -1000000000000.0) {
		tmp = (z / ((x - -1.0) * t_1)) * y;
	} else if (t_2 <= 5e-21) {
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (z / t_1) * (y / (x - -1.0));
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -1000000000000.0) {
		tmp = (z / ((x - -1.0) * t_1)) * y;
	} else if (t_2 <= 5e-21) {
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (z / t_1) * (y / (x - -1.0));
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= -1000000000000.0:
		tmp = (z / ((x - -1.0) * t_1)) * y
	elif t_2 <= 5e-21:
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x)
	elif t_2 <= 2.0:
		tmp = (x - (x / t_1)) / (x + 1.0)
	elif t_2 <= math.inf:
		tmp = (z / t_1) * (y / (x - -1.0))
	else:
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1000000000000.0)
		tmp = Float64(Float64(z / Float64(Float64(x - -1.0) * t_1)) * y);
	elseif (t_2 <= 5e-21)
		tmp = Float64(Float64(-1.0 / Float64(-1.0 - x)) * Float64(Float64(y / t) + x));
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(z / t_1) * Float64(y / Float64(x - -1.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(x - -1.0) / Float64(x + Float64(y / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -1000000000000.0)
		tmp = (z / ((x - -1.0) * t_1)) * y;
	elseif (t_2 <= 5e-21)
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	elseif (t_2 <= 2.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	elseif (t_2 <= Inf)
		tmp = (z / t_1) * (y / (x - -1.0));
	else
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1000000000000.0], N[(N[(z / N[(N[(x - -1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, 5e-21], N[(N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(z / t$95$1), $MachinePrecision] * N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x - -1.0), $MachinePrecision] / N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e12

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot \color{blue}{y} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot \color{blue}{y} \]

      6. lower-/.f64 32.5%

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      8. +-commutative N/A

         \[\leadsto \frac{z}{\left(x + 1\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      9. add-flip N/A

         \[\leadsto \frac{z}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      10. metadata-eval N/A

          \[\leadsto \frac{z}{\left(x - -1\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      11. lower--.f64 32.5%

          \[\leadsto \frac{z}{\left(x - -1\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

   6. Applied rewrites 32.5%

      \[\leadsto \frac{z}{\left(x - -1\right) \cdot \left(t \cdot z - x\right)} \cdot \color{blue}{y} \]

   if -1e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999997e-21

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.4%

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x + \frac{y}{t}}{\color{blue}{x + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y}{t}}{x + 1}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \left(x + \frac{y}{t}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \left(x + \frac{y}{t}\right)} \]

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \cdot \left(x + \frac{y}{t}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \cdot \left(x + \frac{y}{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \cdot \left(x + \frac{y}{t}\right) \]

      9. add-flip N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \left(x + \frac{y}{t}\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(x - \color{blue}{-1}\right)\right)} \cdot \left(x + \frac{y}{t}\right) \]

      11. sub-negate-rev N/A

          \[\leadsto \frac{-1}{\color{blue}{-1 - x}} \cdot \left(x + \frac{y}{t}\right) \]

      12. lower--.f64 70.3%

          \[\leadsto \frac{-1}{\color{blue}{-1 - x}} \cdot \left(x + \frac{y}{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(x + \color{blue}{\frac{y}{t}}\right) \]

      14. +-commutative N/A

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + \color{blue}{x}\right) \]

      15. lower-+.f64 70.3%

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + \color{blue}{x}\right) \]

   8. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + x\right)} \]

   if 4.9999999999999997e-21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{x - \frac{x}{\color{blue}{t \cdot z - x}}}{x + 1} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{x - \frac{x}{t \cdot z - \color{blue}{x}}}{x + 1} \]

      4. lower-*.f64 66.0%

         \[\leadsto \frac{x - \frac{x}{t \cdot z - x}}{x + 1} \]

   4. Applied rewrites 66.0%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. times-frac N/A

         \[\leadsto \frac{y}{1 + x} \cdot \color{blue}{\frac{z}{t \cdot z - x}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{y}{1 + x} \cdot \frac{z}{\color{blue}{t \cdot z - x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{1 + \color{blue}{x}} \]

      9. +-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      11. lower-/.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{\color{blue}{x + 1}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      13. add-flip N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - -1} \]

      15. lower--.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{-1}} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{x - -1}} \]

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      4. lower-unsound-/.f64 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x + 1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      6. add-flip N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      8. metadata-eval 89.3%

         \[\leadsto \frac{1}{\frac{x - \color{blue}{-1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      10. add-flip N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}\right)\right)}} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\mathsf{neg}\left(\color{blue}{\left(y \cdot z - x\right)}\right)}{t \cdot z - x}}} \]

      16. sub-negate-rev N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      17. lower--.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{y \cdot z}}{t \cdot z - x}}} \]

      19. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

      20. lower-*.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

   3. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - -1}{x - \frac{x - z \cdot y}{t \cdot z - x}}}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{x + \color{blue}{\frac{y}{t}}}} \]

      2. lower-/.f64 70.3%

         \[\leadsto \frac{1}{\frac{x - -1}{x + \frac{y}{\color{blue}{t}}}} \]

   6. Applied rewrites 70.3%

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{z}{\left(x - -1\right) \cdot t\_1} \cdot y\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -1000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + x\right)\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - -1}{x + \frac{y}{t}}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (* (/ z (* (- x -1.0) t_1)) y))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_3 -1000000000000.0)
     t_2
     (if (<= t_3 5e-21)
       (* (/ -1.0 (- -1.0 x)) (+ (/ y t) x))
       (if (<= t_3 2.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_3 INFINITY) t_2 (/ 1.0 (/ (- x -1.0) (+ x (/ y t))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (z / ((x - -1.0) * t_1)) * y;
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-21) {
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (z / ((x - -1.0) * t_1)) * y;
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-21) {
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (z / ((x - -1.0) * t_1)) * y
	t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_3 <= -1000000000000.0:
		tmp = t_2
	elif t_3 <= 5e-21:
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x)
	elif t_3 <= 2.0:
		tmp = (x - (x / t_1)) / (x + 1.0)
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(z / Float64(Float64(x - -1.0) * t_1)) * y)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-21)
		tmp = Float64(Float64(-1.0 / Float64(-1.0 - x)) * Float64(Float64(y / t) + x));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(1.0 / Float64(Float64(x - -1.0) / Float64(x + Float64(y / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (z / ((x - -1.0) * t_1)) * y;
	t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-21)
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	elseif (t_3 <= 2.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(N[(x - -1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1000000000000.0], t$95$2, If[LessEqual[t$95$3, 5e-21], N[(N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(1.0 / N[(N[(x - -1.0), $MachinePrecision] / N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e12 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot \color{blue}{y} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot \color{blue}{y} \]

      6. lower-/.f64 32.5%

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      8. +-commutative N/A

         \[\leadsto \frac{z}{\left(x + 1\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      9. add-flip N/A

         \[\leadsto \frac{z}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      10. metadata-eval N/A

          \[\leadsto \frac{z}{\left(x - -1\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      11. lower--.f64 32.5%

          \[\leadsto \frac{z}{\left(x - -1\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

   6. Applied rewrites 32.5%

      \[\leadsto \frac{z}{\left(x - -1\right) \cdot \left(t \cdot z - x\right)} \cdot \color{blue}{y} \]

   if -1e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999997e-21

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.4%

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x + \frac{y}{t}}{\color{blue}{x + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y}{t}}{x + 1}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \left(x + \frac{y}{t}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \left(x + \frac{y}{t}\right)} \]

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \cdot \left(x + \frac{y}{t}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \cdot \left(x + \frac{y}{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \cdot \left(x + \frac{y}{t}\right) \]

      9. add-flip N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \left(x + \frac{y}{t}\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(x - \color{blue}{-1}\right)\right)} \cdot \left(x + \frac{y}{t}\right) \]

      11. sub-negate-rev N/A

          \[\leadsto \frac{-1}{\color{blue}{-1 - x}} \cdot \left(x + \frac{y}{t}\right) \]

      12. lower--.f64 70.3%

          \[\leadsto \frac{-1}{\color{blue}{-1 - x}} \cdot \left(x + \frac{y}{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(x + \color{blue}{\frac{y}{t}}\right) \]

      14. +-commutative N/A

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + \color{blue}{x}\right) \]

      15. lower-+.f64 70.3%

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + \color{blue}{x}\right) \]

   8. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + x\right)} \]

   if 4.9999999999999997e-21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{x - \frac{x}{\color{blue}{t \cdot z - x}}}{x + 1} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{x - \frac{x}{t \cdot z - \color{blue}{x}}}{x + 1} \]

      4. lower-*.f64 66.0%

         \[\leadsto \frac{x - \frac{x}{t \cdot z - x}}{x + 1} \]

   4. Applied rewrites 66.0%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      4. lower-unsound-/.f64 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x + 1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      6. add-flip N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      8. metadata-eval 89.3%

         \[\leadsto \frac{1}{\frac{x - \color{blue}{-1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      10. add-flip N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}\right)\right)}} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\mathsf{neg}\left(\color{blue}{\left(y \cdot z - x\right)}\right)}{t \cdot z - x}}} \]

      16. sub-negate-rev N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      17. lower--.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{y \cdot z}}{t \cdot z - x}}} \]

      19. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

      20. lower-*.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

   3. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - -1}{x - \frac{x - z \cdot y}{t \cdot z - x}}}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{x + \color{blue}{\frac{y}{t}}}} \]

      2. lower-/.f64 70.3%

         \[\leadsto \frac{1}{\frac{x - -1}{x + \frac{y}{\color{blue}{t}}}} \]

   6. Applied rewrites 70.3%

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{z}{\left(x - -1\right) \cdot t\_1} \cdot y\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -1000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;\frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + x\right)\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - -1}{x - -1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - -1}{x + \frac{y}{t}}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (* (/ z (* (- x -1.0) t_1)) y))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_3 -1000000000000.0)
     t_2
     (if (<= t_3 0.2)
       (* (/ -1.0 (- -1.0 x)) (+ (/ y t) x))
       (if (<= t_3 2.0)
         (/ (- x -1.0) (- x -1.0))
         (if (<= t_3 INFINITY) t_2 (/ 1.0 (/ (- x -1.0) (+ x (/ y t))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (z / ((x - -1.0) * t_1)) * y;
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.2) {
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	} else if (t_3 <= 2.0) {
		tmp = (x - -1.0) / (x - -1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (z / ((x - -1.0) * t_1)) * y;
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.2) {
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	} else if (t_3 <= 2.0) {
		tmp = (x - -1.0) / (x - -1.0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (z / ((x - -1.0) * t_1)) * y
	t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_3 <= -1000000000000.0:
		tmp = t_2
	elif t_3 <= 0.2:
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x)
	elif t_3 <= 2.0:
		tmp = (x - -1.0) / (x - -1.0)
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(z / Float64(Float64(x - -1.0) * t_1)) * y)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_3 <= 0.2)
		tmp = Float64(Float64(-1.0 / Float64(-1.0 - x)) * Float64(Float64(y / t) + x));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - -1.0) / Float64(x - -1.0));
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(1.0 / Float64(Float64(x - -1.0) / Float64(x + Float64(y / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (z / ((x - -1.0) * t_1)) * y;
	t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_3 <= 0.2)
		tmp = (-1.0 / (-1.0 - x)) * ((y / t) + x);
	elseif (t_3 <= 2.0)
		tmp = (x - -1.0) / (x - -1.0);
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = 1.0 / ((x - -1.0) / (x + (y / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(N[(x - -1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1000000000000.0], t$95$2, If[LessEqual[t$95$3, 0.2], N[(N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(1.0 / N[(N[(x - -1.0), $MachinePrecision] / N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e12 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot \color{blue}{y} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot \color{blue}{y} \]

      6. lower-/.f64 32.5%

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      8. +-commutative N/A

         \[\leadsto \frac{z}{\left(x + 1\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      9. add-flip N/A

         \[\leadsto \frac{z}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      10. metadata-eval N/A

          \[\leadsto \frac{z}{\left(x - -1\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

      11. lower--.f64 32.5%

          \[\leadsto \frac{z}{\left(x - -1\right) \cdot \left(t \cdot z - x\right)} \cdot y \]

   6. Applied rewrites 32.5%

      \[\leadsto \frac{z}{\left(x - -1\right) \cdot \left(t \cdot z - x\right)} \cdot \color{blue}{y} \]

   if -1e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.4%

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x + \frac{y}{t}}{\color{blue}{x + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y}{t}}{x + 1}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \left(x + \frac{y}{t}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \left(x + \frac{y}{t}\right)} \]

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \cdot \left(x + \frac{y}{t}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \cdot \left(x + \frac{y}{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \cdot \left(x + \frac{y}{t}\right) \]

      9. add-flip N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \left(x + \frac{y}{t}\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(x - \color{blue}{-1}\right)\right)} \cdot \left(x + \frac{y}{t}\right) \]

      11. sub-negate-rev N/A

          \[\leadsto \frac{-1}{\color{blue}{-1 - x}} \cdot \left(x + \frac{y}{t}\right) \]

      12. lower--.f64 70.3%

          \[\leadsto \frac{-1}{\color{blue}{-1 - x}} \cdot \left(x + \frac{y}{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(x + \color{blue}{\frac{y}{t}}\right) \]

      14. +-commutative N/A

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + \color{blue}{x}\right) \]

      15. lower-+.f64 70.3%

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + \color{blue}{x}\right) \]

   8. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + x\right)} \]

   if 0.20000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(1 + \color{blue}{\frac{1}{x}}\right)}{x + 1} \]

      3. lower-/.f64 53.1%

         \[\leadsto \frac{x \cdot \left(1 + \frac{1}{\color{blue}{x}}\right)}{x + 1} \]

   6. Applied rewrites 53.1%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot \color{blue}{x}}{x + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      5. sum-to-mult-rev N/A

         \[\leadsto \frac{x + \color{blue}{1}}{x + 1} \]

      6. add-flip N/A

         \[\leadsto \frac{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{x + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x - -1}{x + 1} \]

      8. lift--.f64 53.1%

         \[\leadsto \frac{x - \color{blue}{-1}}{x + 1} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{x - -1}{x - -1}} \]

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      4. lower-unsound-/.f64 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x + 1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      6. add-flip N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      8. metadata-eval 89.3%

         \[\leadsto \frac{1}{\frac{x - \color{blue}{-1}}{x + \frac{y \cdot z - x}{t \cdot z - x}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}} \]

      10. add-flip N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x - \left(\mathsf{neg}\left(\frac{y \cdot z - x}{t \cdot z - x}\right)\right)}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}\right)\right)}} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \color{blue}{\frac{\mathsf{neg}\left(\left(y \cdot z - x\right)\right)}{t \cdot z - x}}}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\mathsf{neg}\left(\color{blue}{\left(y \cdot z - x\right)}\right)}{t \cdot z - x}}} \]

      16. sub-negate-rev N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      17. lower--.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{\color{blue}{x - y \cdot z}}{t \cdot z - x}}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{y \cdot z}}{t \cdot z - x}}} \]

      19. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

      20. lower-*.f64 89.3%

          \[\leadsto \frac{1}{\frac{x - -1}{x - \frac{x - \color{blue}{z \cdot y}}{t \cdot z - x}}} \]

   3. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - -1}{x - \frac{x - z \cdot y}{t \cdot z - x}}}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{x - -1}{x + \color{blue}{\frac{y}{t}}}} \]

      2. lower-/.f64 70.3%

         \[\leadsto \frac{1}{\frac{x - -1}{x + \frac{y}{\color{blue}{t}}}} \]

   6. Applied rewrites 70.3%

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{x + \frac{y}{t}}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{y}{t} + x\\ t_2 := t \cdot z - x\\ t_3 := \frac{z}{t\_2} \cdot y\\ t_4 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -1000000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 0.2:\\ \;\;\;\;\frac{-1}{-1 - x} \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - -1}{x - -1}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+252}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{x - -1}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y t) x))
        (t_2 (- (* t z) x))
        (t_3 (* (/ z t_2) y))
        (t_4 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_4 -1000000000000.0)
     t_3
     (if (<= t_4 0.2)
       (* (/ -1.0 (- -1.0 x)) t_1)
       (if (<= t_4 2.0)
         (/ (- x -1.0) (- x -1.0))
         (if (<= t_4 2e+252) t_3 (/ t_1 (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / t) + x;
	double t_2 = (t * z) - x;
	double t_3 = (z / t_2) * y;
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -1000000000000.0) {
		tmp = t_3;
	} else if (t_4 <= 0.2) {
		tmp = (-1.0 / (-1.0 - x)) * t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - -1.0) / (x - -1.0);
	} else if (t_4 <= 2e+252) {
		tmp = t_3;
	} else {
		tmp = t_1 / (x - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (y / t) + x
    t_2 = (t * z) - x
    t_3 = (z / t_2) * y
    t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0d0)
    if (t_4 <= (-1000000000000.0d0)) then
        tmp = t_3
    else if (t_4 <= 0.2d0) then
        tmp = ((-1.0d0) / ((-1.0d0) - x)) * t_1
    else if (t_4 <= 2.0d0) then
        tmp = (x - (-1.0d0)) / (x - (-1.0d0))
    else if (t_4 <= 2d+252) then
        tmp = t_3
    else
        tmp = t_1 / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / t) + x;
	double t_2 = (t * z) - x;
	double t_3 = (z / t_2) * y;
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -1000000000000.0) {
		tmp = t_3;
	} else if (t_4 <= 0.2) {
		tmp = (-1.0 / (-1.0 - x)) * t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - -1.0) / (x - -1.0);
	} else if (t_4 <= 2e+252) {
		tmp = t_3;
	} else {
		tmp = t_1 / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / t) + x
	t_2 = (t * z) - x
	t_3 = (z / t_2) * y
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_4 <= -1000000000000.0:
		tmp = t_3
	elif t_4 <= 0.2:
		tmp = (-1.0 / (-1.0 - x)) * t_1
	elif t_4 <= 2.0:
		tmp = (x - -1.0) / (x - -1.0)
	elif t_4 <= 2e+252:
		tmp = t_3
	else:
		tmp = t_1 / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) + x)
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(z / t_2) * y)
	t_4 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -1000000000000.0)
		tmp = t_3;
	elseif (t_4 <= 0.2)
		tmp = Float64(Float64(-1.0 / Float64(-1.0 - x)) * t_1);
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - -1.0) / Float64(x - -1.0));
	elseif (t_4 <= 2e+252)
		tmp = t_3;
	else
		tmp = Float64(t_1 / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / t) + x;
	t_2 = (t * z) - x;
	t_3 = (z / t_2) * y;
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -1000000000000.0)
		tmp = t_3;
	elseif (t_4 <= 0.2)
		tmp = (-1.0 / (-1.0 - x)) * t_1;
	elseif (t_4 <= 2.0)
		tmp = (x - -1.0) / (x - -1.0);
	elseif (t_4 <= 2e+252)
		tmp = t_3;
	else
		tmp = t_1 / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z / t$95$2), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1000000000000.0], t$95$3, If[LessEqual[t$95$4, 0.2], N[(N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(x - -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+252], t$95$3, N[(t$95$1 / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e12 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000002e252

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. times-frac N/A

         \[\leadsto \frac{y}{1 + x} \cdot \color{blue}{\frac{z}{t \cdot z - x}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{y}{1 + x} \cdot \frac{z}{\color{blue}{t \cdot z - x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{1 + \color{blue}{x}} \]

      9. +-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      11. lower-/.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{\color{blue}{x + 1}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      13. add-flip N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - -1} \]

      15. lower--.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{-1}} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{x - -1}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{z}{t \cdot z - x} \cdot y \]
   8. Step-by-step derivation

   9. Applied rewrites 28.7%

      \[\leadsto \frac{z}{t \cdot z - x} \cdot y \]

   if -1e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.4%

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x + \frac{y}{t}}{\color{blue}{x + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y}{t}}{x + 1}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \left(x + \frac{y}{t}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \left(x + \frac{y}{t}\right)} \]

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \cdot \left(x + \frac{y}{t}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \cdot \left(x + \frac{y}{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \cdot \left(x + \frac{y}{t}\right) \]

      9. add-flip N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \left(x + \frac{y}{t}\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(x - \color{blue}{-1}\right)\right)} \cdot \left(x + \frac{y}{t}\right) \]

      11. sub-negate-rev N/A

          \[\leadsto \frac{-1}{\color{blue}{-1 - x}} \cdot \left(x + \frac{y}{t}\right) \]

      12. lower--.f64 70.3%

          \[\leadsto \frac{-1}{\color{blue}{-1 - x}} \cdot \left(x + \frac{y}{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(x + \color{blue}{\frac{y}{t}}\right) \]

      14. +-commutative N/A

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + \color{blue}{x}\right) \]

      15. lower-+.f64 70.3%

          \[\leadsto \frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + \color{blue}{x}\right) \]

   8. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{-1}{-1 - x} \cdot \left(\frac{y}{t} + x\right)} \]

   if 0.20000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(1 + \color{blue}{\frac{1}{x}}\right)}{x + 1} \]

      3. lower-/.f64 53.1%

         \[\leadsto \frac{x \cdot \left(1 + \frac{1}{\color{blue}{x}}\right)}{x + 1} \]

   6. Applied rewrites 53.1%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot \color{blue}{x}}{x + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      5. sum-to-mult-rev N/A

         \[\leadsto \frac{x + \color{blue}{1}}{x + 1} \]

      6. add-flip N/A

         \[\leadsto \frac{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{x + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x - -1}{x + 1} \]

      8. lift--.f64 53.1%

         \[\leadsto \frac{x - \color{blue}{-1}}{x + 1} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{x - -1}{x - -1}} \]

   if 2.0000000000000002e252 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.4%

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x + 1} \]

      3. lower-+.f64 70.4%

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x + 1} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{t} + x}{x - -1}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x - -1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{z}{t\_2} \cdot y\\ t_4 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -1000000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 0.9999999999999999:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - -1}{x - -1}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+252}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (- x -1.0)))
        (t_2 (- (* t z) x))
        (t_3 (* (/ z t_2) y))
        (t_4 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_4 -1000000000000.0)
     t_3
     (if (<= t_4 0.9999999999999999)
       t_1
       (if (<= t_4 2.0)
         (/ (- x -1.0) (- x -1.0))
         (if (<= t_4 2e+252) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (t * z) - x;
	double t_3 = (z / t_2) * y;
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -1000000000000.0) {
		tmp = t_3;
	} else if (t_4 <= 0.9999999999999999) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - -1.0) / (x - -1.0);
	} else if (t_4 <= 2e+252) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = ((y / t) + x) / (x - (-1.0d0))
    t_2 = (t * z) - x
    t_3 = (z / t_2) * y
    t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0d0)
    if (t_4 <= (-1000000000000.0d0)) then
        tmp = t_3
    else if (t_4 <= 0.9999999999999999d0) then
        tmp = t_1
    else if (t_4 <= 2.0d0) then
        tmp = (x - (-1.0d0)) / (x - (-1.0d0))
    else if (t_4 <= 2d+252) then
        tmp = t_3
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (t * z) - x;
	double t_3 = (z / t_2) * y;
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -1000000000000.0) {
		tmp = t_3;
	} else if (t_4 <= 0.9999999999999999) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - -1.0) / (x - -1.0);
	} else if (t_4 <= 2e+252) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x - -1.0)
	t_2 = (t * z) - x
	t_3 = (z / t_2) * y
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_4 <= -1000000000000.0:
		tmp = t_3
	elif t_4 <= 0.9999999999999999:
		tmp = t_1
	elif t_4 <= 2.0:
		tmp = (x - -1.0) / (x - -1.0)
	elif t_4 <= 2e+252:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(z / t_2) * y)
	t_4 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -1000000000000.0)
		tmp = t_3;
	elseif (t_4 <= 0.9999999999999999)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - -1.0) / Float64(x - -1.0));
	elseif (t_4 <= 2e+252)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x - -1.0);
	t_2 = (t * z) - x;
	t_3 = (z / t_2) * y;
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -1000000000000.0)
		tmp = t_3;
	elseif (t_4 <= 0.9999999999999999)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = (x - -1.0) / (x - -1.0);
	elseif (t_4 <= 2e+252)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z / t$95$2), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1000000000000.0], t$95$3, If[LessEqual[t$95$4, 0.9999999999999999], t$95$1, If[LessEqual[t$95$4, 2.0], N[(N[(x - -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+252], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e12 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000002e252

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. times-frac N/A

         \[\leadsto \frac{y}{1 + x} \cdot \color{blue}{\frac{z}{t \cdot z - x}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{y}{1 + x} \cdot \frac{z}{\color{blue}{t \cdot z - x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{1 + \color{blue}{x}} \]

      9. +-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      11. lower-/.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{\color{blue}{x + 1}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      13. add-flip N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - -1} \]

      15. lower--.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{-1}} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{x - -1}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{z}{t \cdot z - x} \cdot y \]
   8. Step-by-step derivation

   9. Applied rewrites 28.7%

      \[\leadsto \frac{z}{t \cdot z - x} \cdot y \]

   if -1e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999999999999989 or 2.0000000000000002e252 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.4%

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x + 1} \]

      3. lower-+.f64 70.4%

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x + 1} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{t} + x}{x - -1}} \]

   if 0.99999999999999989 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(1 + \color{blue}{\frac{1}{x}}\right)}{x + 1} \]

      3. lower-/.f64 53.1%

         \[\leadsto \frac{x \cdot \left(1 + \frac{1}{\color{blue}{x}}\right)}{x + 1} \]

   6. Applied rewrites 53.1%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot \color{blue}{x}}{x + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      5. sum-to-mult-rev N/A

         \[\leadsto \frac{x + \color{blue}{1}}{x + 1} \]

      6. add-flip N/A

         \[\leadsto \frac{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{x + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x - -1}{x + 1} \]

      8. lift--.f64 53.1%

         \[\leadsto \frac{x - \color{blue}{-1}}{x + 1} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{x - -1}{x - -1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{x - -1}{x - -1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{z}{t\_2} \cdot y\\ t_4 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{-47}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-250}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x -1.0) (- x -1.0)))
        (t_2 (- (* t z) x))
        (t_3 (* (/ z t_2) y))
        (t_4 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_4 -1e-47)
     t_3
     (if (<= t_4 2e-250)
       (* (- 1.0 x) x)
       (if (<= t_4 5e-21)
         (/ y t)
         (if (<= t_4 2.0) t_1 (if (<= t_4 INFINITY) t_3 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - -1.0) / (x - -1.0);
	double t_2 = (t * z) - x;
	double t_3 = (z / t_2) * y;
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -1e-47) {
		tmp = t_3;
	} else if (t_4 <= 2e-250) {
		tmp = (1.0 - x) * x;
	} else if (t_4 <= 5e-21) {
		tmp = y / t;
	} else if (t_4 <= 2.0) {
		tmp = t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - -1.0) / (x - -1.0);
	double t_2 = (t * z) - x;
	double t_3 = (z / t_2) * y;
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -1e-47) {
		tmp = t_3;
	} else if (t_4 <= 2e-250) {
		tmp = (1.0 - x) * x;
	} else if (t_4 <= 5e-21) {
		tmp = y / t;
	} else if (t_4 <= 2.0) {
		tmp = t_1;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - -1.0) / (x - -1.0)
	t_2 = (t * z) - x
	t_3 = (z / t_2) * y
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_4 <= -1e-47:
		tmp = t_3
	elif t_4 <= 2e-250:
		tmp = (1.0 - x) * x
	elif t_4 <= 5e-21:
		tmp = y / t
	elif t_4 <= 2.0:
		tmp = t_1
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - -1.0) / Float64(x - -1.0))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(z / t_2) * y)
	t_4 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -1e-47)
		tmp = t_3;
	elseif (t_4 <= 2e-250)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_4 <= 5e-21)
		tmp = Float64(y / t);
	elseif (t_4 <= 2.0)
		tmp = t_1;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - -1.0) / (x - -1.0);
	t_2 = (t * z) - x;
	t_3 = (z / t_2) * y;
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -1e-47)
		tmp = t_3;
	elseif (t_4 <= 2e-250)
		tmp = (1.0 - x) * x;
	elseif (t_4 <= 5e-21)
		tmp = y / t;
	elseif (t_4 <= 2.0)
		tmp = t_1;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z / t$95$2), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1e-47], t$95$3, If[LessEqual[t$95$4, 2e-250], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$4, 5e-21], N[(y / t), $MachinePrecision], If[LessEqual[t$95$4, 2.0], t$95$1, If[LessEqual[t$95$4, Infinity], t$95$3, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999997e-48 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. times-frac N/A

         \[\leadsto \frac{y}{1 + x} \cdot \color{blue}{\frac{z}{t \cdot z - x}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{y}{1 + x} \cdot \frac{z}{\color{blue}{t \cdot z - x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{1 + x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{1 + \color{blue}{x}} \]

      9. +-commutative N/A

         \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      11. lower-/.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{\color{blue}{x + 1}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x + \color{blue}{1}} \]

      13. add-flip N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - -1} \]

      15. lower--.f64 33.4%

          \[\leadsto \frac{z}{t \cdot z - x} \cdot \frac{y}{x - \color{blue}{-1}} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{z}{t \cdot z - x} \cdot \color{blue}{\frac{y}{x - -1}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{z}{t \cdot z - x} \cdot y \]
   8. Step-by-step derivation

   9. Applied rewrites 28.7%

      \[\leadsto \frac{z}{t \cdot z - x} \cdot y \]

   if -9.9999999999999997e-48 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-250

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

      2. lower-+.f64 55.7%

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

   6. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

   7. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-1 \cdot x}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + -1 \cdot \color{blue}{x}\right) \]

      3. lower-*.f64 11.9%

         \[\leadsto x \cdot \left(1 + -1 \cdot x\right) \]

   9. Applied rewrites 11.9%

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto x \cdot \left(1 + \color{blue}{-1 \cdot x}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       3. lower-*.f64 11.9%

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       4. lift-+.f64 N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       5. lift-*.f64 N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       6. mul-1-neg N/A

          \[\leadsto \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \cdot x \]

       7. sub-flip-reverse N/A

          \[\leadsto \left(1 - x\right) \cdot x \]

       8. lower--.f64 11.9%

          \[\leadsto \left(1 - x\right) \cdot x \]

   11. Applied rewrites 11.9%

       \[\leadsto \left(1 - x\right) \cdot x \]

   if 2.0000000000000001e-250 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999997e-21

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 25.0%

         \[\leadsto \frac{y}{\color{blue}{t}} \]

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\frac{y}{t}} \]

   if 4.9999999999999997e-21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(1 + \color{blue}{\frac{1}{x}}\right)}{x + 1} \]

      3. lower-/.f64 53.1%

         \[\leadsto \frac{x \cdot \left(1 + \frac{1}{\color{blue}{x}}\right)}{x + 1} \]

   6. Applied rewrites 53.1%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot \color{blue}{x}}{x + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      5. sum-to-mult-rev N/A

         \[\leadsto \frac{x + \color{blue}{1}}{x + 1} \]

      6. add-flip N/A

         \[\leadsto \frac{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{x + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x - -1}{x + 1} \]

      8. lift--.f64 53.1%

         \[\leadsto \frac{x - \color{blue}{-1}}{x + 1} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{x - -1}{x - -1}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{x - -1}{x - -1}\\ t_2 := \frac{y}{t \cdot \left(1 + x\right)}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-47}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-250}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x -1.0) (- x -1.0)))
        (t_2 (/ y (* t (+ 1.0 x))))
        (t_3 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_3 -1e-47)
     t_2
     (if (<= t_3 2e-250)
       (* (- 1.0 x) x)
       (if (<= t_3 5e-21)
         (/ y t)
         (if (<= t_3 2.0) t_1 (if (<= t_3 INFINITY) t_2 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - -1.0) / (x - -1.0);
	double t_2 = y / (t * (1.0 + x));
	double t_3 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_3 <= -1e-47) {
		tmp = t_2;
	} else if (t_3 <= 2e-250) {
		tmp = (1.0 - x) * x;
	} else if (t_3 <= 5e-21) {
		tmp = y / t;
	} else if (t_3 <= 2.0) {
		tmp = t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - -1.0) / (x - -1.0);
	double t_2 = y / (t * (1.0 + x));
	double t_3 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_3 <= -1e-47) {
		tmp = t_2;
	} else if (t_3 <= 2e-250) {
		tmp = (1.0 - x) * x;
	} else if (t_3 <= 5e-21) {
		tmp = y / t;
	} else if (t_3 <= 2.0) {
		tmp = t_1;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - -1.0) / (x - -1.0)
	t_2 = y / (t * (1.0 + x))
	t_3 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_3 <= -1e-47:
		tmp = t_2
	elif t_3 <= 2e-250:
		tmp = (1.0 - x) * x
	elif t_3 <= 5e-21:
		tmp = y / t
	elif t_3 <= 2.0:
		tmp = t_1
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - -1.0) / Float64(x - -1.0))
	t_2 = Float64(y / Float64(t * Float64(1.0 + x)))
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -1e-47)
		tmp = t_2;
	elseif (t_3 <= 2e-250)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_3 <= 5e-21)
		tmp = Float64(y / t);
	elseif (t_3 <= 2.0)
		tmp = t_1;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - -1.0) / (x - -1.0);
	t_2 = y / (t * (1.0 + x));
	t_3 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -1e-47)
		tmp = t_2;
	elseif (t_3 <= 2e-250)
		tmp = (1.0 - x) * x;
	elseif (t_3 <= 5e-21)
		tmp = y / t;
	elseif (t_3 <= 2.0)
		tmp = t_1;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-47], t$95$2, If[LessEqual[t$95$3, 2e-250], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$3, 5e-21], N[(y / t), $MachinePrecision], If[LessEqual[t$95$3, 2.0], t$95$1, If[LessEqual[t$95$3, Infinity], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999997e-48 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 29.1%

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y}{\color{blue}{t \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{t \cdot \color{blue}{\left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{t \cdot \left(1 + \color{blue}{x}\right)} \]

      3. lower-+.f64 26.9%

         \[\leadsto \frac{y}{t \cdot \left(1 + x\right)} \]

   7. Applied rewrites 26.9%

      \[\leadsto \frac{y}{\color{blue}{t \cdot \left(1 + x\right)}} \]

   if -9.9999999999999997e-48 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-250

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

      2. lower-+.f64 55.7%

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

   6. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

   7. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-1 \cdot x}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + -1 \cdot \color{blue}{x}\right) \]

      3. lower-*.f64 11.9%

         \[\leadsto x \cdot \left(1 + -1 \cdot x\right) \]

   9. Applied rewrites 11.9%

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto x \cdot \left(1 + \color{blue}{-1 \cdot x}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       3. lower-*.f64 11.9%

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       4. lift-+.f64 N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       5. lift-*.f64 N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       6. mul-1-neg N/A

          \[\leadsto \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \cdot x \]

       7. sub-flip-reverse N/A

          \[\leadsto \left(1 - x\right) \cdot x \]

       8. lower--.f64 11.9%

          \[\leadsto \left(1 - x\right) \cdot x \]

   11. Applied rewrites 11.9%

       \[\leadsto \left(1 - x\right) \cdot x \]

   if 2.0000000000000001e-250 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999997e-21

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 25.0%

         \[\leadsto \frac{y}{\color{blue}{t}} \]

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\frac{y}{t}} \]

   if 4.9999999999999997e-21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(1 + \color{blue}{\frac{1}{x}}\right)}{x + 1} \]

      3. lower-/.f64 53.1%

         \[\leadsto \frac{x \cdot \left(1 + \frac{1}{\color{blue}{x}}\right)}{x + 1} \]

   6. Applied rewrites 53.1%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot \color{blue}{x}}{x + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      5. sum-to-mult-rev N/A

         \[\leadsto \frac{x + \color{blue}{1}}{x + 1} \]

      6. add-flip N/A

         \[\leadsto \frac{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{x + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x - -1}{x + 1} \]

      8. lift--.f64 53.1%

         \[\leadsto \frac{x - \color{blue}{-1}}{x + 1} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{x - -1}{x - -1}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-250}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -1}{x - -1}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 -1e-47)
     (/ y t)
     (if (<= t_1 2e-250)
       (* (- 1.0 x) x)
       (if (<= t_1 5e-21) (/ y t) (/ (- x -1.0) (- x -1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e-47) {
		tmp = y / t;
	} else if (t_1 <= 2e-250) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 5e-21) {
		tmp = y / t;
	} else {
		tmp = (x - -1.0) / (x - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_1 <= (-1d-47)) then
        tmp = y / t
    else if (t_1 <= 2d-250) then
        tmp = (1.0d0 - x) * x
    else if (t_1 <= 5d-21) then
        tmp = y / t
    else
        tmp = (x - (-1.0d0)) / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e-47) {
		tmp = y / t;
	} else if (t_1 <= 2e-250) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 5e-21) {
		tmp = y / t;
	} else {
		tmp = (x - -1.0) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -1e-47:
		tmp = y / t
	elif t_1 <= 2e-250:
		tmp = (1.0 - x) * x
	elif t_1 <= 5e-21:
		tmp = y / t
	else:
		tmp = (x - -1.0) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e-47)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-250)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_1 <= 5e-21)
		tmp = Float64(y / t);
	else
		tmp = Float64(Float64(x - -1.0) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -1e-47)
		tmp = y / t;
	elseif (t_1 <= 2e-250)
		tmp = (1.0 - x) * x;
	elseif (t_1 <= 5e-21)
		tmp = y / t;
	else
		tmp = (x - -1.0) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-47], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-250], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 5e-21], N[(y / t), $MachinePrecision], N[(N[(x - -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999997e-48 or 2.0000000000000001e-250 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999997e-21

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 25.0%

         \[\leadsto \frac{y}{\color{blue}{t}} \]

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\frac{y}{t}} \]

   if -9.9999999999999997e-48 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-250

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

      2. lower-+.f64 55.7%

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

   6. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

   7. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-1 \cdot x}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + -1 \cdot \color{blue}{x}\right) \]

      3. lower-*.f64 11.9%

         \[\leadsto x \cdot \left(1 + -1 \cdot x\right) \]

   9. Applied rewrites 11.9%

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto x \cdot \left(1 + \color{blue}{-1 \cdot x}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       3. lower-*.f64 11.9%

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       4. lift-+.f64 N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       5. lift-*.f64 N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       6. mul-1-neg N/A

          \[\leadsto \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \cdot x \]

       7. sub-flip-reverse N/A

          \[\leadsto \left(1 - x\right) \cdot x \]

       8. lower--.f64 11.9%

          \[\leadsto \left(1 - x\right) \cdot x \]

   11. Applied rewrites 11.9%

       \[\leadsto \left(1 - x\right) \cdot x \]

   if 4.9999999999999997e-21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(1 + \color{blue}{\frac{1}{x}}\right)}{x + 1} \]

      3. lower-/.f64 53.1%

         \[\leadsto \frac{x \cdot \left(1 + \frac{1}{\color{blue}{x}}\right)}{x + 1} \]

   6. Applied rewrites 53.1%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{1}{x}\right)}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{1}{x}\right)}}{x + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot \color{blue}{x}}{x + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x}\right) \cdot x}{x + 1} \]

      5. sum-to-mult-rev N/A

         \[\leadsto \frac{x + \color{blue}{1}}{x + 1} \]

      6. add-flip N/A

         \[\leadsto \frac{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{x + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x - -1}{x + 1} \]

      8. lift--.f64 53.1%

         \[\leadsto \frac{x - \color{blue}{-1}}{x + 1} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{x - \color{blue}{-1}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{x - -1}{x - -1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{x}{x - -1}\\ \mathbf{if}\;x \leq -2.36 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (- x -1.0))))
   (if (<= x -2.36e-51) t_1 (if (<= x 1.5e-14) (/ y t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x - -1.0);
	double tmp;
	if (x <= -2.36e-51) {
		tmp = t_1;
	} else if (x <= 1.5e-14) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (x - (-1.0d0))
    if (x <= (-2.36d-51)) then
        tmp = t_1
    else if (x <= 1.5d-14) then
        tmp = y / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (x - -1.0);
	double tmp;
	if (x <= -2.36e-51) {
		tmp = t_1;
	} else if (x <= 1.5e-14) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (x - -1.0)
	tmp = 0
	if x <= -2.36e-51:
		tmp = t_1
	elif x <= 1.5e-14:
		tmp = y / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x - -1.0))
	tmp = 0.0
	if (x <= -2.36e-51)
		tmp = t_1;
	elseif (x <= 1.5e-14)
		tmp = Float64(y / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (x - -1.0);
	tmp = 0.0;
	if (x <= -2.36e-51)
		tmp = t_1;
	elseif (x <= 1.5e-14)
		tmp = y / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.36e-51], t$95$1, If[LessEqual[x, 1.5e-14], N[(y / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3600000000000001e-51 or 1.4999999999999999e-14 < x

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

      2. lower-+.f64 55.7%

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

   6. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{x + \color{blue}{1}} \]

      3. add-flip N/A

         \[\leadsto \frac{x}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x}{x - -1} \]

      5. lift--.f64 55.7%

         \[\leadsto \frac{x}{x - \color{blue}{-1}} \]

   8. Applied rewrites 55.7%

      \[\leadsto \frac{x}{\color{blue}{x - -1}} \]

   if -2.3600000000000001e-51 < x < 1.4999999999999999e-14

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 25.0%

         \[\leadsto \frac{y}{\color{blue}{t}} \]

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 66.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := 1 - \frac{1}{x}\\ \mathbf{if}\;x \leq -0.035:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ 1.0 x))))
   (if (<= x -0.035) t_1 (if (<= x 4.0) (/ y t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (1.0 / x);
	double tmp;
	if (x <= -0.035) {
		tmp = t_1;
	} else if (x <= 4.0) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - (1.0d0 / x)
    if (x <= (-0.035d0)) then
        tmp = t_1
    else if (x <= 4.0d0) then
        tmp = y / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (1.0 / x);
	double tmp;
	if (x <= -0.035) {
		tmp = t_1;
	} else if (x <= 4.0) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - (1.0 / x)
	tmp = 0
	if x <= -0.035:
		tmp = t_1
	elif x <= 4.0:
		tmp = y / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(1.0 / x))
	tmp = 0.0
	if (x <= -0.035)
		tmp = t_1;
	elseif (x <= 4.0)
		tmp = Float64(y / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - (1.0 / x);
	tmp = 0.0;
	if (x <= -0.035)
		tmp = t_1;
	elseif (x <= 4.0)
		tmp = y / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.035], t$95$1, If[LessEqual[x, 4.0], N[(y / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.035000000000000003 or 4 < x

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

      2. lower-+.f64 55.7%

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

   6. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

   7. Taylor expanded in x around inf

      \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \frac{1}{\color{blue}{x}} \]

      2. lower-/.f64 45.4%

         \[\leadsto 1 - \frac{1}{x} \]

   9. Applied rewrites 45.4%

      \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]

   if -0.035000000000000003 < x < 4

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 25.0%

         \[\leadsto \frac{y}{\color{blue}{t}} \]

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 26.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-250}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 -1e-47) (/ y t) (if (<= t_1 2e-250) (* (- 1.0 x) x) (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e-47) {
		tmp = y / t;
	} else if (t_1 <= 2e-250) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_1 <= (-1d-47)) then
        tmp = y / t
    else if (t_1 <= 2d-250) then
        tmp = (1.0d0 - x) * x
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e-47) {
		tmp = y / t;
	} else if (t_1 <= 2e-250) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -1e-47:
		tmp = y / t
	elif t_1 <= 2e-250:
		tmp = (1.0 - x) * x
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e-47)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-250)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -1e-47)
		tmp = y / t;
	elseif (t_1 <= 2e-250)
		tmp = (1.0 - x) * x;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-47], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-250], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], N[(y / t), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999997e-48 or 2.0000000000000001e-250 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 25.0%

         \[\leadsto \frac{y}{\color{blue}{t}} \]

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\frac{y}{t}} \]

   if -9.9999999999999997e-48 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-250

   1. Initial program 89.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x} + x}}{x + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z - x}{t \cdot z - x}} + x}{x + 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z - x}}{t \cdot z - x} + x}{x + 1} \]

      5. sub-flip N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z + \left(\mathsf{neg}\left(x\right)\right)}}{t \cdot z - x} + x}{x + 1} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x}\right)} + x}{x + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t \cdot z - x}} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{t \cdot z - x} \cdot y} + \left(\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}}{x + 1} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t \cdot z - x}}, y, \frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x\right)}{x + 1} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \color{blue}{\frac{\mathsf{neg}\left(x\right)}{t \cdot z - x} + x}\right)}{x + 1} \]

   3. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t \cdot z - x}, y, \frac{x}{x - t \cdot z} + x\right)}}{x + 1} \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

      2. lower-+.f64 55.7%

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

   6. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

   7. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-1 \cdot x}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + -1 \cdot \color{blue}{x}\right) \]

      3. lower-*.f64 11.9%

         \[\leadsto x \cdot \left(1 + -1 \cdot x\right) \]

   9. Applied rewrites 11.9%

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto x \cdot \left(1 + \color{blue}{-1 \cdot x}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       3. lower-*.f64 11.9%

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       4. lift-+.f64 N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       5. lift-*.f64 N/A

          \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]

       6. mul-1-neg N/A

          \[\leadsto \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \cdot x \]

       7. sub-flip-reverse N/A

          \[\leadsto \left(1 - x\right) \cdot x \]

       8. lower--.f64 11.9%

          \[\leadsto \left(1 - x\right) \cdot x \]

   11. Applied rewrites 11.9%

       \[\leadsto \left(1 - x\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 25.0% accurate, 5.6× speedup

Math

\[\frac{y}{t} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ y t))

C

double code(double x, double y, double z, double t) {
	return y / t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y / t
end function

Java

public static double code(double x, double y, double z, double t) {
	return y / t;
}

Python

def code(x, y, z, t):
	return y / t

Julia

function code(x, y, z, t)
	return Float64(y / t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y / t;
end

Wolfram

code[x_, y_, z_, t_] := N[(y / t), $MachinePrecision]

Derivation

1. Initial program 89.4%

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Step-by-step derivation

   1. lower-/.f64 25.0%

      \[\leadsto \frac{y}{\color{blue}{t}} \]

4. Applied rewrites 25.0%

   \[\leadsto \color{blue}{\frac{y}{t}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025191 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))