Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 83.7% → 97.2%

Time: 7.4s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * (y - z)) / (t - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * (y - z)) / (t - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{t - z}{y - z}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / ((t - z) / (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 97.1

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

4. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\frac{x}{\frac{t - z}{y - z}}} \]
5. Add Preprocessing

Alternative 2: 59.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot x\\ t_2 := \frac{t\_1}{t - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{y}{t - z} \cdot x\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-294}:\\ \;\;\;\;\frac{t\_1}{t}\\ \mathbf{elif}\;t\_2 \leq 10^{-108}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{-z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) x)) (t_2 (/ t_1 (- t z))))
   (if (<= t_2 -1e-40)
     (* (/ y (- t z)) x)
     (if (<= t_2 -5e-294)
       (/ t_1 t)
       (if (<= t_2 1e-108) (* (/ z (- z t)) x) (fma x (/ y (- z)) x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * x;
	double t_2 = t_1 / (t - z);
	double tmp;
	if (t_2 <= -1e-40) {
		tmp = (y / (t - z)) * x;
	} else if (t_2 <= -5e-294) {
		tmp = t_1 / t;
	} else if (t_2 <= 1e-108) {
		tmp = (z / (z - t)) * x;
	} else {
		tmp = fma(x, (y / -z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * x)
	t_2 = Float64(t_1 / Float64(t - z))
	tmp = 0.0
	if (t_2 <= -1e-40)
		tmp = Float64(Float64(y / Float64(t - z)) * x);
	elseif (t_2 <= -5e-294)
		tmp = Float64(t_1 / t);
	elseif (t_2 <= 1e-108)
		tmp = Float64(Float64(z / Float64(z - t)) * x);
	else
		tmp = fma(x, Float64(y / Float64(-z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-40], N[(N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, -5e-294], N[(t$95$1 / t), $MachinePrecision], If[LessEqual[t$95$2, 1e-108], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(y / (-z)), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -9.9999999999999993e-41

   1. Initial program 82.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 63.8

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

   7. Applied rewrites 63.8%

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

   if -9.9999999999999993e-41 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -5.0000000000000003e-294

   1. Initial program 95.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 48.1

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

   5. Applied rewrites 48.1%

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

   if -5.0000000000000003e-294 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.00000000000000004e-108

   1. Initial program 90.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.4

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 1.00000000000000004e-108 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 85.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{x \cdot \left(y - z\right)}{z}} \]

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} - \frac{z}{z}\right)} \]

      5. sub-neg N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{z}{z}\right)\right)\right)} \]

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

          \[\leadsto 0 - \left(\frac{x \cdot y}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto 0 - \color{blue}{\left(\frac{x \cdot y}{z} - x\right)} \]

      13. associate-+l- N/A

          \[\leadsto \color{blue}{\left(0 - \frac{x \cdot y}{z}\right) + x} \]

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 60.8%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot x}{t - z} \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{y}{t - z} \cdot x\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot x}{t - z} \leq -5 \cdot 10^{-294}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot x}{t - z} \leq 10^{-108}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{-z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot x\\ t_2 := \frac{t\_1}{t - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-294}:\\ \;\;\;\;\frac{t\_1}{t}\\ \mathbf{elif}\;t\_2 \leq 10^{-108}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{-z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) x)) (t_2 (/ t_1 (- t z))))
   (if (<= t_2 -1e-40)
     (* (/ x (- t z)) y)
     (if (<= t_2 -5e-294)
       (/ t_1 t)
       (if (<= t_2 1e-108) (* (/ z (- z t)) x) (fma x (/ y (- z)) x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * x;
	double t_2 = t_1 / (t - z);
	double tmp;
	if (t_2 <= -1e-40) {
		tmp = (x / (t - z)) * y;
	} else if (t_2 <= -5e-294) {
		tmp = t_1 / t;
	} else if (t_2 <= 1e-108) {
		tmp = (z / (z - t)) * x;
	} else {
		tmp = fma(x, (y / -z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * x)
	t_2 = Float64(t_1 / Float64(t - z))
	tmp = 0.0
	if (t_2 <= -1e-40)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	elseif (t_2 <= -5e-294)
		tmp = Float64(t_1 / t);
	elseif (t_2 <= 1e-108)
		tmp = Float64(Float64(z / Float64(z - t)) * x);
	else
		tmp = fma(x, Float64(y / Float64(-z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-40], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, -5e-294], N[(t$95$1 / t), $MachinePrecision], If[LessEqual[t$95$2, 1e-108], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(y / (-z)), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -9.9999999999999993e-41

   1. Initial program 82.8%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if -9.9999999999999993e-41 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -5.0000000000000003e-294

   1. Initial program 95.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 48.1

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

   5. Applied rewrites 48.1%

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

   if -5.0000000000000003e-294 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.00000000000000004e-108

   1. Initial program 90.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.4

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 1.00000000000000004e-108 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 85.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{x \cdot \left(y - z\right)}{z}} \]

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} - \frac{z}{z}\right)} \]

      5. sub-neg N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{z}{z}\right)\right)\right)} \]

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

          \[\leadsto 0 - \left(\frac{x \cdot y}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto 0 - \color{blue}{\left(\frac{x \cdot y}{z} - x\right)} \]

      13. associate-+l- N/A

          \[\leadsto \color{blue}{\left(0 - \frac{x \cdot y}{z}\right) + x} \]

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 60.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot x}{t - z} \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot x}{t - z} \leq -5 \cdot 10^{-294}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot x}{t - z} \leq 10^{-108}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{-z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot x}{t - z} \leq 10^{-286}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (- y z) x) (- t z)) 1e-286) (* (/ y t) x) (* 1.0 x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((y - z) * x) / (t - z)) <= 1e-286) {
		tmp = (y / t) * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((y - z) * x) / (t - z)) <= 1d-286) then
        tmp = (y / t) * x
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((((y - z) * x) / (t - z)) <= 1e-286) {
		tmp = (y / t) * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (((y - z) * x) / (t - z)) <= 1e-286:
		tmp = (y / t) * x
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(y - z) * x) / Float64(t - z)) <= 1e-286)
		tmp = Float64(Float64(y / t) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((((y - z) * x) / (t - z)) <= 1e-286)
		tmp = (y / t) * x;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], 1e-286], N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.00000000000000005e-286

   1. Initial program 86.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 40.4

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

   5. Applied rewrites 40.4%

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

   7. Applied rewrites 41.1%

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

   if 1.00000000000000005e-286 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 88.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 39.6%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot x}{t - z} \leq 10^{-286}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{z - t} \cdot x\\ \mathbf{if}\;z \leq -9 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{t} \cdot \left(y - z\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (- z t)) x)))
   (if (<= z -9e-42)
     t_1
     (if (<= z 2.2e-131)
       (* (/ x t) (- y z))
       (if (<= z 9.5e-6) (* (/ (- x) z) y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / (z - t)) * x;
	double tmp;
	if (z <= -9e-42) {
		tmp = t_1;
	} else if (z <= 2.2e-131) {
		tmp = (x / t) * (y - z);
	} else if (z <= 9.5e-6) {
		tmp = (-x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z / (z - t)) * x
    if (z <= (-9d-42)) then
        tmp = t_1
    else if (z <= 2.2d-131) then
        tmp = (x / t) * (y - z)
    else if (z <= 9.5d-6) then
        tmp = (-x / z) * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / (z - t)) * x;
	double tmp;
	if (z <= -9e-42) {
		tmp = t_1;
	} else if (z <= 2.2e-131) {
		tmp = (x / t) * (y - z);
	} else if (z <= 9.5e-6) {
		tmp = (-x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / (z - t)) * x
	tmp = 0
	if z <= -9e-42:
		tmp = t_1
	elif z <= 2.2e-131:
		tmp = (x / t) * (y - z)
	elif z <= 9.5e-6:
		tmp = (-x / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / Float64(z - t)) * x)
	tmp = 0.0
	if (z <= -9e-42)
		tmp = t_1;
	elseif (z <= 2.2e-131)
		tmp = Float64(Float64(x / t) * Float64(y - z));
	elseif (z <= 9.5e-6)
		tmp = Float64(Float64(Float64(-x) / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / (z - t)) * x;
	tmp = 0.0;
	if (z <= -9e-42)
		tmp = t_1;
	elseif (z <= 2.2e-131)
		tmp = (x / t) * (y - z);
	elseif (z <= 9.5e-6)
		tmp = (-x / z) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -9e-42], t$95$1, If[LessEqual[z, 2.2e-131], N[(N[(x / t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e-6], N[(N[((-x) / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9e-42 or 9.5000000000000005e-6 < z

   1. Initial program 80.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 73.8

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

   7. Applied rewrites 73.8%

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

   if -9e-42 < z < 2.2e-131

   1. Initial program 93.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 78.8%

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

   if 2.2e-131 < z < 9.5000000000000005e-6

   1. Initial program 97.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{x \cdot \left(y - z\right)}{z}} \]

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} - \frac{z}{z}\right)} \]

      5. sub-neg N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{z}{z}\right)\right)\right)} \]

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

          \[\leadsto 0 - \left(\frac{x \cdot y}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto 0 - \color{blue}{\left(\frac{x \cdot y}{z} - x\right)} \]

      13. associate-+l- N/A

          \[\leadsto \color{blue}{\left(0 - \frac{x \cdot y}{z}\right) + x} \]

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 59.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-42}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{t} \cdot \left(y - z\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{z - t} \cdot x\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (- z t)) x)))
   (if (<= z -6.8e-57)
     t_1
     (if (<= z 1.3e-131)
       (/ (* y x) t)
       (if (<= z 9.5e-6) (* (/ (- x) z) y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / (z - t)) * x;
	double tmp;
	if (z <= -6.8e-57) {
		tmp = t_1;
	} else if (z <= 1.3e-131) {
		tmp = (y * x) / t;
	} else if (z <= 9.5e-6) {
		tmp = (-x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z / (z - t)) * x
    if (z <= (-6.8d-57)) then
        tmp = t_1
    else if (z <= 1.3d-131) then
        tmp = (y * x) / t
    else if (z <= 9.5d-6) then
        tmp = (-x / z) * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / (z - t)) * x;
	double tmp;
	if (z <= -6.8e-57) {
		tmp = t_1;
	} else if (z <= 1.3e-131) {
		tmp = (y * x) / t;
	} else if (z <= 9.5e-6) {
		tmp = (-x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / (z - t)) * x
	tmp = 0
	if z <= -6.8e-57:
		tmp = t_1
	elif z <= 1.3e-131:
		tmp = (y * x) / t
	elif z <= 9.5e-6:
		tmp = (-x / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / Float64(z - t)) * x)
	tmp = 0.0
	if (z <= -6.8e-57)
		tmp = t_1;
	elseif (z <= 1.3e-131)
		tmp = Float64(Float64(y * x) / t);
	elseif (z <= 9.5e-6)
		tmp = Float64(Float64(Float64(-x) / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / (z - t)) * x;
	tmp = 0.0;
	if (z <= -6.8e-57)
		tmp = t_1;
	elseif (z <= 1.3e-131)
		tmp = (y * x) / t;
	elseif (z <= 9.5e-6)
		tmp = (-x / z) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -6.8e-57], t$95$1, If[LessEqual[z, 1.3e-131], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 9.5e-6], N[(N[((-x) / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.80000000000000032e-57 or 9.5000000000000005e-6 < z

   1. Initial program 81.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 73.9

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

   7. Applied rewrites 73.9%

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

   if -6.80000000000000032e-57 < z < 1.29999999999999998e-131

   1. Initial program 92.9%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if 1.29999999999999998e-131 < z < 9.5000000000000005e-6

   1. Initial program 97.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{x \cdot \left(y - z\right)}{z}} \]

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} - \frac{z}{z}\right)} \]

      5. sub-neg N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{z}{z}\right)\right)\right)} \]

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

          \[\leadsto 0 - \left(\frac{x \cdot y}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto 0 - \color{blue}{\left(\frac{x \cdot y}{z} - x\right)} \]

      13. associate-+l- N/A

          \[\leadsto \color{blue}{\left(0 - \frac{x \cdot y}{z}\right) + x} \]

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 59.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\frac{-x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6e-37)
   (fma x (/ t z) x)
   (if (<= z 1.3e-131)
     (/ (* y x) t)
     (if (<= z 2e+24) (* (/ (- x) z) y) (* 1.0 x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6e-37) {
		tmp = fma(x, (t / z), x);
	} else if (z <= 1.3e-131) {
		tmp = (y * x) / t;
	} else if (z <= 2e+24) {
		tmp = (-x / z) * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6e-37)
		tmp = fma(x, Float64(t / z), x);
	elseif (z <= 1.3e-131)
		tmp = Float64(Float64(y * x) / t);
	elseif (z <= 2e+24)
		tmp = Float64(Float64(Float64(-x) / z) * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6e-37], N[(x * N[(t / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.3e-131], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2e+24], N[(N[((-x) / z), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6e-37

   1. Initial program 76.9%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, \frac{t}{\color{blue}{z}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

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

   if -6e-37 < z < 1.29999999999999998e-131

   1. Initial program 93.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   if 1.29999999999999998e-131 < z < 2e24

   1. Initial program 95.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{x \cdot \left(y - z\right)}{z}} \]

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} - \frac{z}{z}\right)} \]

      5. sub-neg N/A

         \[\leadsto 0 - x \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{z}{z}\right)\right)\right)} \]

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

          \[\leadsto 0 - \left(\frac{x \cdot y}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto 0 - \color{blue}{\left(\frac{x \cdot y}{z} - x\right)} \]

      13. associate-+l- N/A

          \[\leadsto \color{blue}{\left(0 - \frac{x \cdot y}{z}\right) + x} \]

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 55.5%

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

   if 2e24 < z

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 68.2%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\frac{-x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{-z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6e-37)
   (fma x (/ t z) x)
   (if (<= z 1.3e-131)
     (/ (* y x) t)
     (if (<= z 2e+24) (* (/ y (- z)) x) (* 1.0 x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6e-37) {
		tmp = fma(x, (t / z), x);
	} else if (z <= 1.3e-131) {
		tmp = (y * x) / t;
	} else if (z <= 2e+24) {
		tmp = (y / -z) * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6e-37)
		tmp = fma(x, Float64(t / z), x);
	elseif (z <= 1.3e-131)
		tmp = Float64(Float64(y * x) / t);
	elseif (z <= 2e+24)
		tmp = Float64(Float64(y / Float64(-z)) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6e-37], N[(x * N[(t / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.3e-131], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2e+24], N[(N[(y / (-z)), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6e-37

   1. Initial program 76.9%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, \frac{t}{\color{blue}{z}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

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

   if -6e-37 < z < 1.29999999999999998e-131

   1. Initial program 93.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   if 1.29999999999999998e-131 < z < 2e24

   1. Initial program 95.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, \frac{t}{\color{blue}{z}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 13.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 53.0%

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

   if 2e24 < z

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 68.2%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{-z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{z - t} \cdot x\\ \mathbf{if}\;z \leq -4.35 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (- z t)) x)))
   (if (<= z -4.35e+136)
     t_1
     (if (<= z 4.9e+182) (* (/ x (- t z)) (- y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / (z - t)) * x;
	double tmp;
	if (z <= -4.35e+136) {
		tmp = t_1;
	} else if (z <= 4.9e+182) {
		tmp = (x / (t - z)) * (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z / (z - t)) * x
    if (z <= (-4.35d+136)) then
        tmp = t_1
    else if (z <= 4.9d+182) then
        tmp = (x / (t - z)) * (y - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / (z - t)) * x;
	double tmp;
	if (z <= -4.35e+136) {
		tmp = t_1;
	} else if (z <= 4.9e+182) {
		tmp = (x / (t - z)) * (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / (z - t)) * x
	tmp = 0
	if z <= -4.35e+136:
		tmp = t_1
	elif z <= 4.9e+182:
		tmp = (x / (t - z)) * (y - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / Float64(z - t)) * x)
	tmp = 0.0
	if (z <= -4.35e+136)
		tmp = t_1;
	elseif (z <= 4.9e+182)
		tmp = Float64(Float64(x / Float64(t - z)) * Float64(y - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / (z - t)) * x;
	tmp = 0.0;
	if (z <= -4.35e+136)
		tmp = t_1;
	elseif (z <= 4.9e+182)
		tmp = (x / (t - z)) * (y - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -4.35e+136], t$95$1, If[LessEqual[z, 4.9e+182], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.34999999999999987e136 or 4.9e182 < z

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 95.3

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

   7. Applied rewrites 95.3%

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

   if -4.34999999999999987e136 < z < 4.9e182

   1. Initial program 92.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 94.8

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

   4. Applied rewrites 94.8%

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

Alternative 10: 72.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z} \cdot y\\ \mathbf{if}\;y \leq -11500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x (- t z)) y)))
   (if (<= y -11500.0) t_1 (if (<= y 2.55e-48) (* (/ z (- z t)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / (t - z)) * y;
	double tmp;
	if (y <= -11500.0) {
		tmp = t_1;
	} else if (y <= 2.55e-48) {
		tmp = (z / (z - t)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / (t - z)) * y
    if (y <= (-11500.0d0)) then
        tmp = t_1
    else if (y <= 2.55d-48) then
        tmp = (z / (z - t)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / (t - z)) * y;
	double tmp;
	if (y <= -11500.0) {
		tmp = t_1;
	} else if (y <= 2.55e-48) {
		tmp = (z / (z - t)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / (t - z)) * y
	tmp = 0
	if y <= -11500.0:
		tmp = t_1
	elif y <= 2.55e-48:
		tmp = (z / (z - t)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(t - z)) * y)
	tmp = 0.0
	if (y <= -11500.0)
		tmp = t_1;
	elseif (y <= 2.55e-48)
		tmp = Float64(Float64(z / Float64(z - t)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / (t - z)) * y;
	tmp = 0.0;
	if (y <= -11500.0)
		tmp = t_1;
	elseif (y <= 2.55e-48)
		tmp = (z / (z - t)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -11500.0], t$95$1, If[LessEqual[y, 2.55e-48], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -11500 or 2.55000000000000006e-48 < y

   1. Initial program 89.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -11500 < y < 2.55000000000000006e-48

   1. Initial program 85.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 80.7

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

   7. Applied rewrites 80.7%

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

Alternative 11: 60.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6e-37)
   (fma x (/ t z) x)
   (if (<= z 6.2e+70) (/ (* y x) t) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6e-37) {
		tmp = fma(x, (t / z), x);
	} else if (z <= 6.2e+70) {
		tmp = (y * x) / t;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6e-37)
		tmp = fma(x, Float64(t / z), x);
	elseif (z <= 6.2e+70)
		tmp = Float64(Float64(y * x) / t);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6e-37], N[(x * N[(t / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 6.2e+70], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6e-37

   1. Initial program 76.9%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, \frac{t}{\color{blue}{z}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

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

   if -6e-37 < z < 6.2000000000000006e70

   1. Initial program 94.2%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.5

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

   5. Applied rewrites 56.5%

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

   if 6.2000000000000006e70 < z

   1. Initial program 78.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 76.0%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-34}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.6e-34) (* 1.0 x) (if (<= z 6.2e+70) (/ (* y x) t) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.6e-34) {
		tmp = 1.0 * x;
	} else if (z <= 6.2e+70) {
		tmp = (y * x) / t;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-4.6d-34)) then
        tmp = 1.0d0 * x
    else if (z <= 6.2d+70) then
        tmp = (y * x) / t
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.6e-34) {
		tmp = 1.0 * x;
	} else if (z <= 6.2e+70) {
		tmp = (y * x) / t;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4.6e-34:
		tmp = 1.0 * x
	elif z <= 6.2e+70:
		tmp = (y * x) / t
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.6e-34)
		tmp = Float64(1.0 * x);
	elseif (z <= 6.2e+70)
		tmp = Float64(Float64(y * x) / t);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4.6e-34)
		tmp = 1.0 * x;
	elseif (z <= 6.2e+70)
		tmp = (y * x) / t;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.6e-34], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 6.2e+70], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.60000000000000022e-34 or 6.2000000000000006e70 < z

   1. Initial program 77.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 63.7%

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

   if -4.60000000000000022e-34 < z < 6.2000000000000006e70

   1. Initial program 94.3%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

Alternative 13: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-41}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.5e-41) (* 1.0 x) (if (<= z 5e+70) (* (/ x t) y) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.5e-41) {
		tmp = 1.0 * x;
	} else if (z <= 5e+70) {
		tmp = (x / t) * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.5d-41)) then
        tmp = 1.0d0 * x
    else if (z <= 5d+70) then
        tmp = (x / t) * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.5e-41) {
		tmp = 1.0 * x;
	} else if (z <= 5e+70) {
		tmp = (x / t) * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.5e-41:
		tmp = 1.0 * x
	elif z <= 5e+70:
		tmp = (x / t) * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.5e-41)
		tmp = Float64(1.0 * x);
	elseif (z <= 5e+70)
		tmp = Float64(Float64(x / t) * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.5e-41)
		tmp = 1.0 * x;
	elseif (z <= 5e+70)
		tmp = (x / t) * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.5e-41], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 5e+70], N[(N[(x / t), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.49999999999999994e-41 or 5.0000000000000002e70 < z

   1. Initial program 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 62.7%

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

   if -1.49999999999999994e-41 < z < 5.0000000000000002e70

   1. Initial program 94.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.9

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

   5. Applied rewrites 56.9%

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

   7. Applied rewrites 56.9%

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

Alternative 14: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{y - z}{t - z} \cdot x \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((y - z) / (t - z)) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\frac{y - z}{t - z} \cdot x} \]
5. Add Preprocessing

Alternative 15: 35.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in z around inf

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

7. Applied rewrites 33.9%

   \[\leadsto \color{blue}{1} \cdot x \]
8. Add Preprocessing

Developer Target 1: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{t - z}{y - z}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / ((t - z) / (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ x (/ (- t z) (- y z))))

  (/ (* x (- y z)) (- t z)))