Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Percentage Accurate: 68.2% → 90.8%

Time: 6.4s

Alternatives: 16

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

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 - x)) / (a - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - x)) / (a - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{x - t}{z - a}, x\right)\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-266}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(1, t, \frac{y - a}{z} \cdot x\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+308}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- x t) (- z a)) x))
        (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e-266)
       t_2
       (if (<= t_2 0.0)
         (fma 1.0 t (* (/ (- y a) z) x))
         (if (<= t_2 1e+308) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), ((x - t) / (z - a)), x);
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e-266) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = fma(1.0, t, (((y - a) / z) * x));
	} else if (t_2 <= 1e+308) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(Float64(x - t) / Float64(z - a)), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e-266)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = fma(1.0, t, Float64(Float64(Float64(y - a) / z) * x));
	elseif (t_2 <= 1e+308)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e-266], t$95$2, If[LessEqual[t$95$2, 0.0], N[(1.0 * t + N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+308], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -inf.0 or 1e308 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 38.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. negate-sub2 N/A

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

      14. negate-sub2 N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 82.2

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

   3. Applied rewrites 82.2%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -9.9999999999999998e-267 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 1e308

   1. Initial program 96.5%

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

   if -9.9999999999999998e-267 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 10.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{z - a}, t, \frac{y - a}{z} \cdot x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 93.6

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(1, t, \frac{y - a}{z} \cdot x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 93.1%

       \[\leadsto \mathsf{fma}\left(1, t, \frac{y - a}{z} \cdot x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{z - a}, t, \frac{y - a}{z} \cdot x\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{+164}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{x - t}{z - a}, x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+79}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) (- z a)) t (* (/ (- y a) z) x))))
   (if (<= z -4e+164)
     t_1
     (if (<= z -7e-211)
       (fma (- y z) (/ (- x t) (- z a)) x)
       (if (<= z 3.6e+79) (+ x (/ (* (- y z) (- t x)) (- a z))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / (z - a)), t, (((y - a) / z) * x));
	double tmp;
	if (z <= -4e+164) {
		tmp = t_1;
	} else if (z <= -7e-211) {
		tmp = fma((y - z), ((x - t) / (z - a)), x);
	} else if (z <= 3.6e+79) {
		tmp = x + (((y - z) * (t - x)) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / Float64(z - a)), t, Float64(Float64(Float64(y - a) / z) * x))
	tmp = 0.0
	if (z <= -4e+164)
		tmp = t_1;
	elseif (z <= -7e-211)
		tmp = fma(Float64(y - z), Float64(Float64(x - t) / Float64(z - a)), x);
	elseif (z <= 3.6e+79)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * t + N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4e+164], t$95$1, If[LessEqual[z, -7e-211], N[(N[(y - z), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3.6e+79], N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4e164 or 3.5999999999999999e79 < z

   1. Initial program 35.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{z - a}, t, \frac{y - a}{z} \cdot x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -4e164 < z < -7e-211

   1. Initial program 76.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. negate-sub2 N/A

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

      14. negate-sub2 N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 85.8

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

   3. Applied rewrites 85.8%

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

   if -7e-211 < z < 3.5999999999999999e79

   1. Initial program 86.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - y}{z - a}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t, \frac{y - a}{z} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t, \left(\frac{y - z}{z - a} + 1\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z y) (- z a))))
   (if (<= z -1.6e+194)
     (fma t_1 t (* (/ (- y a) z) x))
     (fma t_1 t (* (+ (/ (- y z) (- z a)) 1.0) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - y) / (z - a);
	double tmp;
	if (z <= -1.6e+194) {
		tmp = fma(t_1, t, (((y - a) / z) * x));
	} else {
		tmp = fma(t_1, t, ((((y - z) / (z - a)) + 1.0) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - y) / Float64(z - a))
	tmp = 0.0
	if (z <= -1.6e+194)
		tmp = fma(t_1, t, Float64(Float64(Float64(y - a) / z) * x));
	else
		tmp = fma(t_1, t, Float64(Float64(Float64(Float64(y - z) / Float64(z - a)) + 1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.6e+194], N[(t$95$1 * t + N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t + N[(N[(N[(N[(y - z), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000011e194

   1. Initial program 26.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{z - a}, t, \frac{y - a}{z} \cdot x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 92.3

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

   7. Applied rewrites 92.3%

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

   if -1.60000000000000011e194 < z

   1. Initial program 72.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 88.8%

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

Alternative 4: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, t, \frac{y - a}{z} \cdot x\right)\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{x - t}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 t (* (/ (- y a) z) x))))
   (if (<= z -1.45e+168)
     t_1
     (if (<= z 2.5e+171) (fma (- y z) (/ (- x t) (- z a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, t, (((y - a) / z) * x));
	double tmp;
	if (z <= -1.45e+168) {
		tmp = t_1;
	} else if (z <= 2.5e+171) {
		tmp = fma((y - z), ((x - t) / (z - a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, t, Float64(Float64(Float64(y - a) / z) * x))
	tmp = 0.0
	if (z <= -1.45e+168)
		tmp = t_1;
	elseif (z <= 2.5e+171)
		tmp = fma(Float64(y - z), Float64(Float64(x - t) / Float64(z - a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * t + N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e+168], t$95$1, If[LessEqual[z, 2.5e+171], N[(N[(y - z), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e168 or 2.5000000000000002e171 < z

   1. Initial program 29.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{z - a}, t, \frac{y - a}{z} \cdot x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 91.2

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

   7. Applied rewrites 91.2%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(1, t, \frac{y - a}{z} \cdot x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 81.0%

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

   if -1.45e168 < z < 2.5000000000000002e171

   1. Initial program 79.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. negate-sub2 N/A

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

      14. negate-sub2 N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 87.1

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

   3. Applied rewrites 87.1%

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

Alternative 5: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z - a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- t x) (- a z)))))
   (if (<= y -1.55e+66)
     t_1
     (if (<= y 6.4e+105) (fma (/ (- z y) (- z a)) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((t - x) / (a - z));
	double tmp;
	if (y <= -1.55e+66) {
		tmp = t_1;
	} else if (y <= 6.4e+105) {
		tmp = fma(((z - 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(t - x) / Float64(a - z)))
	tmp = 0.0
	if (y <= -1.55e+66)
		tmp = t_1;
	elseif (y <= 6.4e+105)
		tmp = fma(Float64(Float64(z - y) / 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[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.55e+66], t$95$1, If[LessEqual[y, 6.4e+105], N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55000000000000009e66 or 6.4e105 < y

   1. Initial program 68.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. flip-- N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto x + \frac{\left(y \cdot y - \color{blue}{{z}^{2}}\right) \cdot \left(t - x\right)}{\left(y + z\right) \cdot \left(a - z\right)} \]

      12. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{\left(y \cdot y - {z}^{2}\right)} \cdot \left(t - x\right)}{\left(y + z\right) \cdot \left(a - z\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto x + \frac{\left(\color{blue}{y \cdot y} - {z}^{2}\right) \cdot \left(t - x\right)}{\left(y + z\right) \cdot \left(a - z\right)} \]

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lift--.f64 32.3

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

   3. Applied rewrites 32.3%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. sub-div N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 75.2

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

   6. Applied rewrites 75.2%

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

   if -1.55000000000000009e66 < y < 6.4e105

   1. Initial program 68.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{z - a}, t, \frac{y - a}{z} \cdot x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 58.8

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

   7. Applied rewrites 58.8%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{z - a}, t, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 71.2%

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

Alternative 6: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{if}\;a \leq -1.12 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.78 \cdot 10^{+37}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ (- y z) a) x)))
   (if (<= a -1.12e+18)
     t_1
     (if (<= a 1.78e+37) (* t (/ (- y z) (- a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), ((y - z) / a), x);
	double tmp;
	if (a <= -1.12e+18) {
		tmp = t_1;
	} else if (a <= 1.78e+37) {
		tmp = t * ((y - z) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(Float64(y - z) / a), x)
	tmp = 0.0
	if (a <= -1.12e+18)
		tmp = t_1;
	elseif (a <= 1.78e+37)
		tmp = Float64(t * Float64(Float64(y - z) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.12e+18], t$95$1, If[LessEqual[a, 1.78e+37], N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.12e18 or 1.78e37 < a

   1. Initial program 68.9%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 77.3

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

   4. Applied rewrites 77.3%

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

   if -1.12e18 < a < 1.78e37

   1. Initial program 67.7%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-div N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 61.9

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

   7. Applied rewrites 61.9%

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

Alternative 7: 64.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(t - x\right) \cdot y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ (- y z) (- a z)))))
   (if (<= z -2.05e-63)
     t_1
     (if (<= z 8.5e-59)
       (fma y (/ (- t x) a) x)
       (if (<= z 4.6e+79) (/ (* (- t x) y) (- a z)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * ((y - z) / (a - z));
	double tmp;
	if (z <= -2.05e-63) {
		tmp = t_1;
	} else if (z <= 8.5e-59) {
		tmp = fma(y, ((t - x) / a), x);
	} else if (z <= 4.6e+79) {
		tmp = ((t - x) * y) / (a - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(Float64(y - z) / Float64(a - z)))
	tmp = 0.0
	if (z <= -2.05e-63)
		tmp = t_1;
	elseif (z <= 8.5e-59)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	elseif (z <= 4.6e+79)
		tmp = Float64(Float64(Float64(t - x) * y) / Float64(a - z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.05e-63], t$95$1, If[LessEqual[z, 8.5e-59], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.6e+79], N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.0499999999999999e-63 or 4.6000000000000001e79 < z

   1. Initial program 46.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 84.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-div N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 60.0

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

   7. Applied rewrites 60.0%

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

   if -2.0499999999999999e-63 < z < 8.49999999999999933e-59

   1. Initial program 90.4%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 77.0

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

   4. Applied rewrites 77.0%

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

   if 8.49999999999999933e-59 < z < 4.6000000000000001e79

   1. Initial program 77.5%

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

   2. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 39.4

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

   4. Applied rewrites 39.4%

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

Alternative 8: 64.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+109}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- t x) a) x)))
   (if (<= a -1.3e+18)
     t_1
     (if (<= a 1.6e+109) (* t (/ (- y z) (- a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((t - x) / a), x);
	double tmp;
	if (a <= -1.3e+18) {
		tmp = t_1;
	} else if (a <= 1.6e+109) {
		tmp = t * ((y - z) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(t - x) / a), x)
	tmp = 0.0
	if (a <= -1.3e+18)
		tmp = t_1;
	elseif (a <= 1.6e+109)
		tmp = Float64(t * Float64(Float64(y - z) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.3e+18], t$95$1, If[LessEqual[a, 1.6e+109], N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3e18 or 1.6000000000000001e109 < a

   1. Initial program 68.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 69.4

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

   4. Applied rewrites 69.4%

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

   if -1.3e18 < a < 1.6000000000000001e109

   1. Initial program 68.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-div N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 60.5

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

   7. Applied rewrites 60.5%

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

Alternative 9: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ t (* y (/ x z)))))
   (if (<= z -1.45e+30) t_1 (if (<= z 8.5e-59) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t + (y * (x / z));
	double tmp;
	if (z <= -1.45e+30) {
		tmp = t_1;
	} else if (z <= 8.5e-59) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t + Float64(y * Float64(x / z)))
	tmp = 0.0
	if (z <= -1.45e+30)
		tmp = t_1;
	elseif (z <= 8.5e-59)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t + N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e+30], t$95$1, If[LessEqual[z, 8.5e-59], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4499999999999999e30 or 8.49999999999999933e-59 < z

   1. Initial program 48.7%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in a around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-*.f64 43.6

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

   7. Applied rewrites 43.6%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 63.6

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

   10. Applied rewrites 63.6%

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

   11. Taylor expanded in x around inf

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

       1. lift-/.f64 55.5

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

   13. Applied rewrites 55.5%

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

   if -1.4499999999999999e30 < z < 8.49999999999999933e-59

   1. Initial program 89.3%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 73.7

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

   4. Applied rewrites 73.7%

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

Alternative 10: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ t (* y (/ x z)))))
   (if (<= z -8.5e+16) t_1 (if (<= z 8.5e-59) (fma y (/ t a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t + (y * (x / z));
	double tmp;
	if (z <= -8.5e+16) {
		tmp = t_1;
	} else if (z <= 8.5e-59) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t + Float64(y * Float64(x / z)))
	tmp = 0.0
	if (z <= -8.5e+16)
		tmp = t_1;
	elseif (z <= 8.5e-59)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t + N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.5e+16], t$95$1, If[LessEqual[z, 8.5e-59], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e16 or 8.49999999999999933e-59 < z

   1. Initial program 49.1%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in a around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-*.f64 43.6

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

   7. Applied rewrites 43.6%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 63.4

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

   10. Applied rewrites 63.4%

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

   11. Taylor expanded in x around inf

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

       1. lift-/.f64 55.2

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

   13. Applied rewrites 55.2%

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

   if -8.5e16 < z < 8.49999999999999933e-59

   1. Initial program 89.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 74.1

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{t}{a}, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 60.8%

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

Alternative 11: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+63}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+72}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.6e+63)
   t
   (if (<= z 3.2e-51) (fma y (/ t a) x) (if (<= z 9e+72) (/ (* x y) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e+63) {
		tmp = t;
	} else if (z <= 3.2e-51) {
		tmp = fma(y, (t / a), x);
	} else if (z <= 9e+72) {
		tmp = (x * y) / z;
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.6e+63)
		tmp = t;
	elseif (z <= 3.2e-51)
		tmp = fma(y, Float64(t / a), x);
	elseif (z <= 9e+72)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.6e+63], t, If[LessEqual[z, 3.2e-51], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 9e+72], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6000000000000001e63 or 8.9999999999999997e72 < z

   1. Initial program 39.2%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   3. Step-by-step derivation

   4. Applied rewrites 47.7%

      \[\leadsto \color{blue}{t} \]

   if -2.6000000000000001e63 < z < 3.2e-51

   1. Initial program 88.1%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{t}{a}, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 59.3%

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

   if 3.2e-51 < z < 8.9999999999999997e72

   1. Initial program 77.4%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in a around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-*.f64 45.9

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

   7. Applied rewrites 45.9%

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

   8. Taylor expanded in x around inf

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

      1. lift-*.f64 19.0

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

   10. Applied rewrites 19.0%

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

Alternative 12: 49.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;a \leq -6 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-80}:\\ \;\;\;\;\frac{t \cdot \left(z - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ t a) x)))
   (if (<= a -6e-46) t_1 (if (<= a 3.1e-80) (/ (* t (- z y)) z) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (t / a), x);
	double tmp;
	if (a <= -6e-46) {
		tmp = t_1;
	} else if (a <= 3.1e-80) {
		tmp = (t * (z - y)) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(t / a), x)
	tmp = 0.0
	if (a <= -6e-46)
		tmp = t_1;
	elseif (a <= 3.1e-80)
		tmp = Float64(Float64(t * Float64(z - y)) / z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -6e-46], t$95$1, If[LessEqual[a, 3.1e-80], N[(N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.99999999999999975e-46 or 3.10000000000000016e-80 < a

   1. Initial program 69.0%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{t}{a}, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 54.0%

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

   if -5.99999999999999975e-46 < a < 3.10000000000000016e-80

   1. Initial program 67.1%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in a around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-*.f64 65.3

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

   7. Applied rewrites 65.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{t \cdot \left(z - y\right)}{z} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift--.f64 43.8

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

   10. Applied rewrites 43.8%

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

Alternative 13: 38.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+187}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -4.5e+146)
   (/ (* t y) a)
   (if (<= y 3.3e+187) (+ x t) (/ (* x y) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -4.5e+146) {
		tmp = (t * y) / a;
	} else if (y <= 3.3e+187) {
		tmp = x + t;
	} else {
		tmp = (x * y) / z;
	}
	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) :: tmp
    if (y <= (-4.5d+146)) then
        tmp = (t * y) / a
    else if (y <= 3.3d+187) then
        tmp = x + t
    else
        tmp = (x * y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -4.5e+146) {
		tmp = (t * y) / a;
	} else if (y <= 3.3e+187) {
		tmp = x + t;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -4.5e+146:
		tmp = (t * y) / a
	elif y <= 3.3e+187:
		tmp = x + t
	else:
		tmp = (x * y) / z
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -4.5e+146)
		tmp = Float64(Float64(t * y) / a);
	elseif (y <= 3.3e+187)
		tmp = Float64(x + t);
	else
		tmp = Float64(Float64(x * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -4.5e+146)
		tmp = (t * y) / a;
	elseif (y <= 3.3e+187)
		tmp = x + t;
	else
		tmp = (x * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -4.5e+146], N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[y, 3.3e+187], N[(x + t), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.50000000000000026e146

   1. Initial program 68.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 59.1

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 29.9

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

   7. Applied rewrites 29.9%

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

   if -4.50000000000000026e146 < y < 3.3000000000000001e187

   1. Initial program 68.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. flip-- N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto x + \frac{\left(y \cdot y - \color{blue}{{z}^{2}}\right) \cdot \left(t - x\right)}{\left(y + z\right) \cdot \left(a - z\right)} \]

      12. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{\left(y \cdot y - {z}^{2}\right)} \cdot \left(t - x\right)}{\left(y + z\right) \cdot \left(a - z\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto x + \frac{\left(\color{blue}{y \cdot y} - {z}^{2}\right) \cdot \left(t - x\right)}{\left(y + z\right) \cdot \left(a - z\right)} \]

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lift--.f64 49.8

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

   3. Applied rewrites 49.8%

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

   4. Taylor expanded in z around inf

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

      1. lift--.f64 22.1

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

   6. Applied rewrites 22.1%

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

   7. Taylor expanded in x around 0

      \[\leadsto x + t \]
   8. Step-by-step derivation

   9. Applied rewrites 39.8%

      \[\leadsto x + t \]

   if 3.3000000000000001e187 < y

   1. Initial program 68.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. negate-sub2 N/A

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

      8. negate-sub2 N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in a around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-*.f64 42.1

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in x around inf

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

      1. lift-*.f64 27.5

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

   10. Applied rewrites 27.5%

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

Alternative 14: 37.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot y}{a}\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+102}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* t y) a)))
   (if (<= y -4.5e+146) t_1 (if (<= y 2.8e+102) (+ x t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t * y) / a;
	double tmp;
	if (y <= -4.5e+146) {
		tmp = t_1;
	} else if (y <= 2.8e+102) {
		tmp = x + 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 = (t * y) / a
    if (y <= (-4.5d+146)) then
        tmp = t_1
    else if (y <= 2.8d+102) then
        tmp = x + 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 = (t * y) / a;
	double tmp;
	if (y <= -4.5e+146) {
		tmp = t_1;
	} else if (y <= 2.8e+102) {
		tmp = x + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (t * y) / a
	tmp = 0
	if y <= -4.5e+146:
		tmp = t_1
	elif y <= 2.8e+102:
		tmp = x + t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t * y) / a)
	tmp = 0.0
	if (y <= -4.5e+146)
		tmp = t_1;
	elseif (y <= 2.8e+102)
		tmp = Float64(x + t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (t * y) / a;
	tmp = 0.0;
	if (y <= -4.5e+146)
		tmp = t_1;
	elseif (y <= 2.8e+102)
		tmp = x + t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[y, -4.5e+146], t$95$1, If[LessEqual[y, 2.8e+102], N[(x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000026e146 or 2.80000000000000018e102 < y

   1. Initial program 68.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 28.0

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

   7. Applied rewrites 28.0%

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

   if -4.50000000000000026e146 < y < 2.80000000000000018e102

   1. Initial program 68.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. flip-- N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto x + \frac{\left(y \cdot y - \color{blue}{{z}^{2}}\right) \cdot \left(t - x\right)}{\left(y + z\right) \cdot \left(a - z\right)} \]

      12. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{\left(y \cdot y - {z}^{2}\right)} \cdot \left(t - x\right)}{\left(y + z\right) \cdot \left(a - z\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto x + \frac{\left(\color{blue}{y \cdot y} - {z}^{2}\right) \cdot \left(t - x\right)}{\left(y + z\right) \cdot \left(a - z\right)} \]

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lift--.f64 51.2

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

   3. Applied rewrites 51.2%

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

   4. Taylor expanded in z around inf

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

      1. lift--.f64 22.8

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

   6. Applied rewrites 22.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto x + t \]
   8. Step-by-step derivation

   9. Applied rewrites 41.0%

      \[\leadsto x + t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 37.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+93}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.3e+130) x (if (<= a 1.25e+93) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.3e+130) {
		tmp = x;
	} else if (a <= 1.25e+93) {
		tmp = t;
	} else {
		tmp = x;
	}
	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) :: tmp
    if (a <= (-4.3d+130)) then
        tmp = x
    else if (a <= 1.25d+93) then
        tmp = t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.3e+130) {
		tmp = x;
	} else if (a <= 1.25e+93) {
		tmp = t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -4.3e+130:
		tmp = x
	elif a <= 1.25e+93:
		tmp = t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.3e+130)
		tmp = x;
	elseif (a <= 1.25e+93)
		tmp = t;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -4.3e+130)
		tmp = x;
	elseif (a <= 1.25e+93)
		tmp = t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.3e+130], x, If[LessEqual[a, 1.25e+93], t, x]]

Derivation

1. Split input into 2 regimes

2. if a < -4.29999999999999984e130 or 1.25e93 < a

   1. Initial program 68.5%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 53.2%

      \[\leadsto \color{blue}{x} \]

   if -4.29999999999999984e130 < a < 1.25e93

   1. Initial program 68.1%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   3. Step-by-step derivation

   4. Applied rewrites 31.1%

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

Alternative 16: 24.6% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return 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 = t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 68.2%

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

2. Taylor expanded in z around inf

   \[\leadsto \color{blue}{t} \]
3. Step-by-step derivation

4. Applied rewrites 24.6%

   \[\leadsto \color{blue}{t} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y z) (- t x)) (- a z))))