Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Percentage Accurate: 77.6% → 92.1%

Time: 4.0s

Alternatives: 14

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (x + y) - (((z - t) * y) / (a - 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, a)
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), intent (in) :: a
    code = (x + y) - (((z - t) * y) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (x + y) - (((z - t) * y) / (a - 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, a)
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), intent (in) :: a
    code = (x + y) - (((z - t) * y) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{t - a}, \mathsf{fma}\left(y, \frac{t}{a - t}, y + x\right)\right)\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-202}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y (- t a)) (fma y (/ t (- a t)) (+ y x))))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (<= t_2 -5e-202) t_1 (if (<= t_2 5e-252) (fma y (/ (- z a) t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / (t - a)), fma(y, (t / (a - t)), (y + x)));
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -5e-202) {
		tmp = t_1;
	} else if (t_2 <= 5e-252) {
		tmp = fma(y, ((z - a) / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / Float64(t - a)), fma(y, Float64(t / Float64(a - t)), Float64(y + x)))
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -5e-202)
		tmp = t_1;
	elseif (t_2 <= 5e-252)
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y * N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-202], t$95$1, If[LessEqual[t$95$2, 5e-252], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -4.99999999999999973e-202 or 5.00000000000000008e-252 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-out N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-rgt-in N/A

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

      12. div-add N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. associate-+l+ N/A

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

   3. Applied rewrites 87.5%

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

   if -4.99999999999999973e-202 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 5.00000000000000008e-252

   1. Initial program 77.6%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-2neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-out-- N/A

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

      16. associate-/l* N/A

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   6. Applied rewrites 59.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{z - a}{t}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-118}:\\ \;\;\;\;x - \left(\frac{y}{t - a} \cdot \left(t - z\right) - y\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-153}:\\ \;\;\;\;x + -1 \cdot \frac{a \cdot y - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a - t}, y, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5.8e-118)
   (- x (- (* (/ y (- t a)) (- t z)) y))
   (if (<= a 3.7e-153)
     (+ x (* -1.0 (/ (- (* a y) (* y z)) t)))
     (fma (/ (- t z) (- a t)) y (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5.8e-118) {
		tmp = x - (((y / (t - a)) * (t - z)) - y);
	} else if (a <= 3.7e-153) {
		tmp = x + (-1.0 * (((a * y) - (y * z)) / t));
	} else {
		tmp = fma(((t - z) / (a - t)), y, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5.8e-118)
		tmp = Float64(x - Float64(Float64(Float64(y / Float64(t - a)) * Float64(t - z)) - y));
	elseif (a <= 3.7e-153)
		tmp = Float64(x + Float64(-1.0 * Float64(Float64(Float64(a * y) - Float64(y * z)) / t)));
	else
		tmp = fma(Float64(Float64(t - z) / Float64(a - t)), y, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5.8e-118], N[(x - N[(N[(N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.7e-153], N[(x + N[(-1.0 * N[(N[(N[(a * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.79999999999999961e-118

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 80.7

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. distribute-lft-neg-out N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. distribute-neg-frac2 N/A

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

      18. lower-/.f64 N/A

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

      19. lift--.f64 N/A

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

      20. sub-negate-rev N/A

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

      21. lower--.f64 N/A

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

      22. lower--.f64 87.5

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

   3. Applied rewrites 87.5%

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

   if -5.79999999999999961e-118 < a < 3.7000000000000003e-153

   1. Initial program 77.6%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

   if 3.7000000000000003e-153 < a

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y + x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-118}:\\ \;\;\;\;x - \left(\frac{y}{t - a} \cdot \left(t - z\right) - y\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t}, \left(z - a\right) \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a - t}, y, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5.8e-118)
   (- x (- (* (/ y (- t a)) (- t z)) y))
   (if (<= a 3.7e-153)
     (fma (/ 1.0 t) (* (- z a) y) x)
     (fma (/ (- t z) (- a t)) y (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5.8e-118) {
		tmp = x - (((y / (t - a)) * (t - z)) - y);
	} else if (a <= 3.7e-153) {
		tmp = fma((1.0 / t), ((z - a) * y), x);
	} else {
		tmp = fma(((t - z) / (a - t)), y, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5.8e-118)
		tmp = Float64(x - Float64(Float64(Float64(y / Float64(t - a)) * Float64(t - z)) - y));
	elseif (a <= 3.7e-153)
		tmp = fma(Float64(1.0 / t), Float64(Float64(z - a) * y), x);
	else
		tmp = fma(Float64(Float64(t - z) / Float64(a - t)), y, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5.8e-118], N[(x - N[(N[(N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.7e-153], N[(N[(1.0 / t), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.79999999999999961e-118

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 80.7

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. distribute-lft-neg-out N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. distribute-neg-frac2 N/A

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

      18. lower-/.f64 N/A

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

      19. lift--.f64 N/A

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

      20. sub-negate-rev N/A

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

      21. lower--.f64 N/A

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

      22. lower--.f64 87.5

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

   3. Applied rewrites 87.5%

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

   if -5.79999999999999961e-118 < a < 3.7000000000000003e-153

   1. Initial program 77.6%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. distribute-lft-neg-in N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out-- N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. lower--.f64 57.4

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

   6. Applied rewrites 57.4%

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

   if 3.7000000000000003e-153 < a

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y + x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, t - z, y\right) + x\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a - t}, y, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.9e+22)
   (+ (fma (/ y (- a t)) (- t z) y) x)
   (if (<= a 3.1e-153)
     (fma y (/ (- z a) t) x)
     (fma (/ (- t z) (- a t)) y (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.9e+22) {
		tmp = fma((y / (a - t)), (t - z), y) + x;
	} else if (a <= 3.1e-153) {
		tmp = fma(y, ((z - a) / t), x);
	} else {
		tmp = fma(((t - z) / (a - t)), y, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.9e+22)
		tmp = Float64(fma(Float64(y / Float64(a - t)), Float64(t - z), y) + x);
	elseif (a <= 3.1e-153)
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	else
		tmp = fma(Float64(Float64(t - z) / Float64(a - t)), y, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.9e+22], N[(N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(t - z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 3.1e-153], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.9000000000000002e22

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. sub-flip N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 83.4

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

   3. Applied rewrites 83.4%

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

   if -1.9000000000000002e22 < a < 3.09999999999999995e-153

   1. Initial program 77.6%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-2neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-out-- N/A

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

      16. associate-/l* N/A

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   6. Applied rewrites 59.9%

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

   if 3.09999999999999995e-153 < a

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y + x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a - t}, t - z, y\right) + x\\ \mathbf{if}\;a \leq -1.9 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (fma (/ y (- a t)) (- t z) y) x)))
   (if (<= a -1.9e+22) t_1 (if (<= a 3.1e-153) (fma y (/ (- z a) t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / (a - t)), (t - z), y) + x;
	double tmp;
	if (a <= -1.9e+22) {
		tmp = t_1;
	} else if (a <= 3.1e-153) {
		tmp = fma(y, ((z - a) / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(fma(Float64(y / Float64(a - t)), Float64(t - z), y) + x)
	tmp = 0.0
	if (a <= -1.9e+22)
		tmp = t_1;
	elseif (a <= 3.1e-153)
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(t - z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.9e+22], t$95$1, If[LessEqual[a, 3.1e-153], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9000000000000002e22 or 3.09999999999999995e-153 < a

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. sub-flip N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 83.4

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

   3. Applied rewrites 83.4%

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

   if -1.9000000000000002e22 < a < 3.09999999999999995e-153

   1. Initial program 77.6%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-2neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-out-- N/A

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

      16. associate-/l* N/A

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   6. Applied rewrites 59.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{z - a}{t}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \left(\frac{z}{a} \cdot y - x\right)\\ \mathbf{if}\;a \leq -3.15 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (- (* (/ z a) y) x))))
   (if (<= a -3.15e+22) t_1 (if (<= a 5.5e-50) (fma y (/ (- z a) t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (((z / a) * y) - x);
	double tmp;
	if (a <= -3.15e+22) {
		tmp = t_1;
	} else if (a <= 5.5e-50) {
		tmp = fma(y, ((z - a) / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(Float64(Float64(z / a) * y) - x))
	tmp = 0.0
	if (a <= -3.15e+22)
		tmp = t_1;
	elseif (a <= 5.5e-50)
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.15e+22], t$95$1, If[LessEqual[a, 5.5e-50], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.1500000000000001e22 or 5.49999999999999975e-50 < a

   1. Initial program 77.6%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 65.8

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

   4. Applied rewrites 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 67.6

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

   6. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 67.6

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

   8. Applied rewrites 67.6%

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

   if -3.1500000000000001e22 < a < 5.49999999999999975e-50

   1. Initial program 77.6%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-2neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-out-- N/A

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

      16. associate-/l* N/A

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   6. Applied rewrites 59.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{z - a}{t}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - -1 \cdot x\\ \mathbf{if}\;a \leq -1.66 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.05 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (* -1.0 x))))
   (if (<= a -1.66e+25) t_1 (if (<= a 4.05e-47) (fma y (/ (- z a) t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (-1.0 * x);
	double tmp;
	if (a <= -1.66e+25) {
		tmp = t_1;
	} else if (a <= 4.05e-47) {
		tmp = fma(y, ((z - a) / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(-1.0 * x))
	tmp = 0.0
	if (a <= -1.66e+25)
		tmp = t_1;
	elseif (a <= 4.05e-47)
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.66e+25], t$95$1, If[LessEqual[a, 4.05e-47], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6600000000000001e25 or 4.0500000000000002e-47 < a

   1. Initial program 77.6%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 65.8

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

   4. Applied rewrites 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 67.6

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

   6. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto y - \color{blue}{-1 \cdot x} \]
   10. Step-by-step derivation

       1. lower-*.f64 60.9

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

   11. Applied rewrites 60.9%

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

   if -1.6600000000000001e25 < a < 4.0500000000000002e-47

   1. Initial program 77.6%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-2neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-out-- N/A

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

      16. associate-/l* N/A

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   6. Applied rewrites 59.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{z - a}{t}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - -1 \cdot x\\ \mathbf{if}\;a \leq -9 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-47}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (* -1.0 x))))
   (if (<= a -9e+22) t_1 (if (<= a 3.7e-47) (+ x (/ (* y z) t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (-1.0 * x);
	double tmp;
	if (a <= -9e+22) {
		tmp = t_1;
	} else if (a <= 3.7e-47) {
		tmp = x + ((y * z) / 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, a)
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), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y - ((-1.0d0) * x)
    if (a <= (-9d+22)) then
        tmp = t_1
    else if (a <= 3.7d-47) then
        tmp = x + ((y * z) / 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 a) {
	double t_1 = y - (-1.0 * x);
	double tmp;
	if (a <= -9e+22) {
		tmp = t_1;
	} else if (a <= 3.7e-47) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y - (-1.0 * x)
	tmp = 0
	if a <= -9e+22:
		tmp = t_1
	elif a <= 3.7e-47:
		tmp = x + ((y * z) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(-1.0 * x))
	tmp = 0.0
	if (a <= -9e+22)
		tmp = t_1;
	elseif (a <= 3.7e-47)
		tmp = Float64(x + Float64(Float64(y * z) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y - (-1.0 * x);
	tmp = 0.0;
	if (a <= -9e+22)
		tmp = t_1;
	elseif (a <= 3.7e-47)
		tmp = x + ((y * z) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9e+22], t$95$1, If[LessEqual[a, 3.7e-47], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.9999999999999996e22 or 3.7e-47 < a

   1. Initial program 77.6%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 65.8

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

   4. Applied rewrites 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 67.6

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

   6. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto y - \color{blue}{-1 \cdot x} \]
   10. Step-by-step derivation

       1. lower-*.f64 60.9

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

   11. Applied rewrites 60.9%

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

   if -8.9999999999999996e22 < a < 3.7e-47

   1. Initial program 77.6%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 60.5

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

   7. Applied rewrites 60.5%

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

Alternative 9: 69.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t - a} \cdot z\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ t_3 := y - -1 \cdot x\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-146}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-252}:\\ \;\;\;\;x + \left(y + -1 \cdot y\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+270}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- t a)) z))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t))))
        (t_3 (- y (* -1.0 x))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e-146)
       t_3
       (if (<= t_2 5e-252)
         (+ x (+ y (* -1.0 y)))
         (if (<= t_2 2e+270) t_3 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (t - a)) * z;
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double t_3 = y - (-1.0 * x);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e-146) {
		tmp = t_3;
	} else if (t_2 <= 5e-252) {
		tmp = x + (y + (-1.0 * y));
	} else if (t_2 <= 2e+270) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (t - a)) * z;
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double t_3 = y - (-1.0 * x);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -1e-146) {
		tmp = t_3;
	} else if (t_2 <= 5e-252) {
		tmp = x + (y + (-1.0 * y));
	} else if (t_2 <= 2e+270) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / (t - a)) * z
	t_2 = (x + y) - (((z - t) * y) / (a - t))
	t_3 = y - (-1.0 * x)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -1e-146:
		tmp = t_3
	elif t_2 <= 5e-252:
		tmp = x + (y + (-1.0 * y))
	elif t_2 <= 2e+270:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(t - a)) * z)
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	t_3 = Float64(y - Float64(-1.0 * x))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e-146)
		tmp = t_3;
	elseif (t_2 <= 5e-252)
		tmp = Float64(x + Float64(y + Float64(-1.0 * y)));
	elseif (t_2 <= 2e+270)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / (t - a)) * z;
	t_2 = (x + y) - (((z - t) * y) / (a - t));
	t_3 = y - (-1.0 * x);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -1e-146)
		tmp = t_3;
	elseif (t_2 <= 5e-252)
		tmp = x + (y + (-1.0 * y));
	elseif (t_2 <= 2e+270)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y - N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e-146], t$95$3, If[LessEqual[t$95$2, 5e-252], N[(x + N[(y + N[(-1.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+270], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0 or 2.0000000000000001e270 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-out N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-rgt-in N/A

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

      12. div-add N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. associate-+l+ N/A

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

   3. Applied rewrites 87.5%

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

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{t - a}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 26.4

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

   6. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\frac{y \cdot z}{t - a}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 28.3

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

   8. Applied rewrites 28.3%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -1.00000000000000003e-146 or 5.00000000000000008e-252 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 2.0000000000000001e270

   1. Initial program 77.6%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 65.8

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

   4. Applied rewrites 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 67.6

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

   6. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto y - \color{blue}{-1 \cdot x} \]
   10. Step-by-step derivation

       1. lower-*.f64 60.9

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

   11. Applied rewrites 60.9%

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

   if -1.00000000000000003e-146 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 5.00000000000000008e-252

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-out N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-rgt-in N/A

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

      12. div-add N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. associate-+l+ N/A

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

   3. Applied rewrites 87.5%

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

   4. Taylor expanded in t around inf

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 51.3

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

   6. Applied rewrites 51.3%

      \[\leadsto \color{blue}{x + \left(y + -1 \cdot y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - -1 \cdot x\\ \mathbf{if}\;a \leq -6 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;x + \left(y + -1 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (* -1.0 x))))
   (if (<= a -6e+21) t_1 (if (<= a 1.4e-43) (+ x (+ y (* -1.0 y))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (-1.0 * x);
	double tmp;
	if (a <= -6e+21) {
		tmp = t_1;
	} else if (a <= 1.4e-43) {
		tmp = x + (y + (-1.0 * y));
	} 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, a)
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), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y - ((-1.0d0) * x)
    if (a <= (-6d+21)) then
        tmp = t_1
    else if (a <= 1.4d-43) then
        tmp = x + (y + ((-1.0d0) * y))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (-1.0 * x);
	double tmp;
	if (a <= -6e+21) {
		tmp = t_1;
	} else if (a <= 1.4e-43) {
		tmp = x + (y + (-1.0 * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y - (-1.0 * x)
	tmp = 0
	if a <= -6e+21:
		tmp = t_1
	elif a <= 1.4e-43:
		tmp = x + (y + (-1.0 * y))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(-1.0 * x))
	tmp = 0.0
	if (a <= -6e+21)
		tmp = t_1;
	elseif (a <= 1.4e-43)
		tmp = Float64(x + Float64(y + Float64(-1.0 * y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y - (-1.0 * x);
	tmp = 0.0;
	if (a <= -6e+21)
		tmp = t_1;
	elseif (a <= 1.4e-43)
		tmp = x + (y + (-1.0 * y));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6e+21], t$95$1, If[LessEqual[a, 1.4e-43], N[(x + N[(y + N[(-1.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6e21 or 1.3999999999999999e-43 < a

   1. Initial program 77.6%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 65.8

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

   4. Applied rewrites 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 67.6

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

   6. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto y - \color{blue}{-1 \cdot x} \]
   10. Step-by-step derivation

       1. lower-*.f64 60.9

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

   11. Applied rewrites 60.9%

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

   if -6e21 < a < 1.3999999999999999e-43

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-out N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-rgt-in N/A

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

      12. div-add N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. associate-+l+ N/A

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

   3. Applied rewrites 87.5%

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

   4. Taylor expanded in t around inf

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 51.3

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

   6. Applied rewrites 51.3%

      \[\leadsto \color{blue}{x + \left(y + -1 \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - -1 \cdot x\\ \mathbf{if}\;a \leq -1.36 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{-210}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (* -1.0 x))))
   (if (<= a -1.36e-49) t_1 (if (<= a 2.05e-210) (/ (* y z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (-1.0 * x);
	double tmp;
	if (a <= -1.36e-49) {
		tmp = t_1;
	} else if (a <= 2.05e-210) {
		tmp = (y * z) / 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, a)
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), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y - ((-1.0d0) * x)
    if (a <= (-1.36d-49)) then
        tmp = t_1
    else if (a <= 2.05d-210) then
        tmp = (y * z) / 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 a) {
	double t_1 = y - (-1.0 * x);
	double tmp;
	if (a <= -1.36e-49) {
		tmp = t_1;
	} else if (a <= 2.05e-210) {
		tmp = (y * z) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y - (-1.0 * x)
	tmp = 0
	if a <= -1.36e-49:
		tmp = t_1
	elif a <= 2.05e-210:
		tmp = (y * z) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(-1.0 * x))
	tmp = 0.0
	if (a <= -1.36e-49)
		tmp = t_1;
	elseif (a <= 2.05e-210)
		tmp = Float64(Float64(y * z) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y - (-1.0 * x);
	tmp = 0.0;
	if (a <= -1.36e-49)
		tmp = t_1;
	elseif (a <= 2.05e-210)
		tmp = (y * z) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.36e-49], t$95$1, If[LessEqual[a, 2.05e-210], N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.36000000000000006e-49 or 2.04999999999999995e-210 < a

   1. Initial program 77.6%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 65.8

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

   4. Applied rewrites 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 67.6

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

   6. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto y - \color{blue}{-1 \cdot x} \]
   10. Step-by-step derivation

       1. lower-*.f64 60.9

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

   11. Applied rewrites 60.9%

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

   if -1.36000000000000006e-49 < a < 2.04999999999999995e-210

   1. Initial program 77.6%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 18.2

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

   7. Applied rewrites 18.2%

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 59.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - -1 \cdot x\\ \mathbf{if}\;a \leq -1.36 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-209}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (* -1.0 x))))
   (if (<= a -1.36e-49) t_1 (if (<= a 5.8e-209) (* (/ y t) z) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (-1.0 * x);
	double tmp;
	if (a <= -1.36e-49) {
		tmp = t_1;
	} else if (a <= 5.8e-209) {
		tmp = (y / t) * z;
	} 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, a)
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), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y - ((-1.0d0) * x)
    if (a <= (-1.36d-49)) then
        tmp = t_1
    else if (a <= 5.8d-209) then
        tmp = (y / t) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (-1.0 * x);
	double tmp;
	if (a <= -1.36e-49) {
		tmp = t_1;
	} else if (a <= 5.8e-209) {
		tmp = (y / t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y - (-1.0 * x)
	tmp = 0
	if a <= -1.36e-49:
		tmp = t_1
	elif a <= 5.8e-209:
		tmp = (y / t) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(-1.0 * x))
	tmp = 0.0
	if (a <= -1.36e-49)
		tmp = t_1;
	elseif (a <= 5.8e-209)
		tmp = Float64(Float64(y / t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y - (-1.0 * x);
	tmp = 0.0;
	if (a <= -1.36e-49)
		tmp = t_1;
	elseif (a <= 5.8e-209)
		tmp = (y / t) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.36e-49], t$95$1, If[LessEqual[a, 5.8e-209], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.36000000000000006e-49 or 5.80000000000000052e-209 < a

   1. Initial program 77.6%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 65.8

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

   4. Applied rewrites 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 67.6

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

   6. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto y - \color{blue}{-1 \cdot x} \]
   10. Step-by-step derivation

       1. lower-*.f64 60.9

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

   11. Applied rewrites 60.9%

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

   if -1.36000000000000006e-49 < a < 5.80000000000000052e-209

   1. Initial program 77.6%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 18.2

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

   7. Applied rewrites 18.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 19.5

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

   9. Applied rewrites 19.5%

      \[\leadsto \frac{y}{t} \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ y - -1 \cdot x \end{array} \]

FPCore

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

C

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

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, a)
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), intent (in) :: a
    code = y - ((-1.0d0) * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 65.8

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

4. Applied rewrites 65.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 67.6

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

6. Applied rewrites 67.6%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-+.f64 N/A

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

   4. associate--r+ N/A

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

   5. sub-negate N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 67.6

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

8. Applied rewrites 67.6%

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

9. Taylor expanded in x around inf

   \[\leadsto y - \color{blue}{-1 \cdot x} \]
10. Step-by-step derivation

    1. lower-*.f64 60.9

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

11. Applied rewrites 60.9%

    \[\leadsto y - \color{blue}{-1 \cdot x} \]
12. Add Preprocessing

Alternative 14: 45.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ y - \left(y - x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- y (- y x)))

C

double code(double x, double y, double z, double t, double a) {
	return y - (y - x);
}

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, a)
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), intent (in) :: a
    code = y - (y - x)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return y - (y - x);
}

Python

def code(x, y, z, t, a):
	return y - (y - x)

Julia

function code(x, y, z, t, a)
	return Float64(y - Float64(y - x))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = y - (y - x);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(y - N[(y - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 65.8

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

4. Applied rewrites 65.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 67.6

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

6. Applied rewrites 67.6%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-+.f64 N/A

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

   4. associate--r+ N/A

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

   5. sub-negate N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 67.6

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

8. Applied rewrites 67.6%

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

9. Taylor expanded in t around inf

   \[\leadsto y - \color{blue}{\left(y - x\right)} \]
10. Step-by-step derivation

    1. lower--.f64 45.4

       \[\leadsto y - \left(y - \color{blue}{x}\right) \]

11. Applied rewrites 45.4%

    \[\leadsto y - \color{blue}{\left(y - x\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64
  (- (+ x y) (/ (* (- z t) y) (- a t))))