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

Percentage Accurate: 83.7% → 97.2%

Time: 7.5s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 97.5

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

   6. Applied rewrites 97.5%

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

   7. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 63.8

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

   9. Applied rewrites 63.8%

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

   11. Applied rewrites 63.8%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 96.3

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 75.2

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

   9. 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. associate-/l* N/A

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

      16. *-commutative N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 60.8%

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

Alternative 3: 58.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{y}{t - z} \cdot x\\ \mathbf{elif}\;t\_1 \leq 10^{-286}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;t\_1 \leq 10^{-108}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-z}, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))))
   (if (<= t_1 -1e-40)
     (* (/ y (- t z)) x)
     (if (<= t_1 1e-286)
       (/ (* (- y z) x) t)
       (if (<= t_1 1e-108) (/ (* z x) (- z t)) (fma (/ y (- z)) x x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -1e-40) {
		tmp = (y / (t - z)) * x;
	} else if (t_1 <= 1e-286) {
		tmp = ((y - z) * x) / t;
	} else if (t_1 <= 1e-108) {
		tmp = (z * x) / (z - t);
	} else {
		tmp = fma((y / -z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= -1e-40)
		tmp = Float64(Float64(y / Float64(t - z)) * x);
	elseif (t_1 <= 1e-286)
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	elseif (t_1 <= 1e-108)
		tmp = Float64(Float64(z * x) / Float64(z - t));
	else
		tmp = fma(Float64(y / Float64(-z)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-40], N[(N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 1e-286], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 1e-108], N[(N[(z * x), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(N[(y / (-z)), $MachinePrecision] * x + 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. 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 97.5

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

   6. Applied rewrites 97.5%

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

   7. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 63.8

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

   9. Applied rewrites 63.8%

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

   11. Applied rewrites 63.8%

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

   if -9.9999999999999993e-41 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.00000000000000005e-286

   1. Initial program 90.6%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   if 1.00000000000000005e-286 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.00000000000000004e-108

   1. Initial program 99.6%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 96.1

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

   4. Applied rewrites 96.1%

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

   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 69.6

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

   7. Applied rewrites 69.6%

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

   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. associate-/l* N/A

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

      16. *-commutative N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 60.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{y}{t - z} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 10^{-286}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 10^{-108}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-z}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{-286}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;t\_1 \leq 10^{-108}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-z}, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))))
   (if (<= t_1 -1e-40)
     (* (/ x (- t z)) y)
     (if (<= t_1 1e-286)
       (/ (* (- y z) x) t)
       (if (<= t_1 1e-108) (/ (* z x) (- z t)) (fma (/ y (- z)) x x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -1e-40) {
		tmp = (x / (t - z)) * y;
	} else if (t_1 <= 1e-286) {
		tmp = ((y - z) * x) / t;
	} else if (t_1 <= 1e-108) {
		tmp = (z * x) / (z - t);
	} else {
		tmp = fma((y / -z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= -1e-40)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	elseif (t_1 <= 1e-286)
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	elseif (t_1 <= 1e-108)
		tmp = Float64(Float64(z * x) / Float64(z - t));
	else
		tmp = fma(Float64(y / Float64(-z)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-40], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 1e-286], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 1e-108], N[(N[(z * x), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(N[(y / (-z)), $MachinePrecision] * x + 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)) < 1.00000000000000005e-286

   1. Initial program 90.6%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   if 1.00000000000000005e-286 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.00000000000000004e-108

   1. Initial program 99.6%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 96.1

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

   4. Applied rewrites 96.1%

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

   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 69.6

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

   7. Applied rewrites 69.6%

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

   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. associate-/l* N/A

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

      16. *-commutative N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 60.8%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 10^{-286}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 10^{-108}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-z}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], 1e-286], N[(x * N[(y / t), $MachinePrecision]), $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. 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 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 96.0

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 39.6%

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

Alternative 6: 66.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot x}{z - t}\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+213}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z x) (- z t))))
   (if (<= z -7.4e+213)
     (* 1.0 x)
     (if (<= z -1.5e-33)
       t_1
       (if (<= z -6.8e-142)
         (/ (* (- y z) x) t)
         (if (<= z 1.05e+52) (* (/ x (- t z)) y) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * x) / (z - t);
	double tmp;
	if (z <= -7.4e+213) {
		tmp = 1.0 * x;
	} else if (z <= -1.5e-33) {
		tmp = t_1;
	} else if (z <= -6.8e-142) {
		tmp = ((y - z) * x) / t;
	} else if (z <= 1.05e+52) {
		tmp = (x / (t - 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 * x) / (z - t)
    if (z <= (-7.4d+213)) then
        tmp = 1.0d0 * x
    else if (z <= (-1.5d-33)) then
        tmp = t_1
    else if (z <= (-6.8d-142)) then
        tmp = ((y - z) * x) / t
    else if (z <= 1.05d+52) then
        tmp = (x / (t - 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 * x) / (z - t);
	double tmp;
	if (z <= -7.4e+213) {
		tmp = 1.0 * x;
	} else if (z <= -1.5e-33) {
		tmp = t_1;
	} else if (z <= -6.8e-142) {
		tmp = ((y - z) * x) / t;
	} else if (z <= 1.05e+52) {
		tmp = (x / (t - z)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * x) / (z - t)
	tmp = 0
	if z <= -7.4e+213:
		tmp = 1.0 * x
	elif z <= -1.5e-33:
		tmp = t_1
	elif z <= -6.8e-142:
		tmp = ((y - z) * x) / t
	elif z <= 1.05e+52:
		tmp = (x / (t - z)) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * x) / Float64(z - t))
	tmp = 0.0
	if (z <= -7.4e+213)
		tmp = Float64(1.0 * x);
	elseif (z <= -1.5e-33)
		tmp = t_1;
	elseif (z <= -6.8e-142)
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	elseif (z <= 1.05e+52)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * x) / (z - t);
	tmp = 0.0;
	if (z <= -7.4e+213)
		tmp = 1.0 * x;
	elseif (z <= -1.5e-33)
		tmp = t_1;
	elseif (z <= -6.8e-142)
		tmp = ((y - z) * x) / t;
	elseif (z <= 1.05e+52)
		tmp = (x / (t - z)) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.4e+213], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, -1.5e-33], t$95$1, If[LessEqual[z, -6.8e-142], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.05e+52], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.39999999999999985e213

   1. Initial program 42.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. 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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 94.0%

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

   if -7.39999999999999985e213 < z < -1.5000000000000001e-33 or 1.05e52 < z

   1. Initial program 85.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. 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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 71.3

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

   7. Applied rewrites 71.3%

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

   if -1.5000000000000001e-33 < z < -6.80000000000000057e-142

   1. Initial program 99.7%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

   if -6.80000000000000057e-142 < z < 1.05e52

   1. Initial program 92.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 78.5

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

   5. Applied rewrites 78.5%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+213}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot x}{z - t}\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+213}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z x) (- z t))))
   (if (<= z -7.4e+213)
     (* 1.0 x)
     (if (<= z -8e-40)
       t_1
       (if (<= z -4e-141)
         (* x (/ (- y z) t))
         (if (<= z 1.05e+52) (* (/ x (- t z)) y) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * x) / (z - t);
	double tmp;
	if (z <= -7.4e+213) {
		tmp = 1.0 * x;
	} else if (z <= -8e-40) {
		tmp = t_1;
	} else if (z <= -4e-141) {
		tmp = x * ((y - z) / t);
	} else if (z <= 1.05e+52) {
		tmp = (x / (t - 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 * x) / (z - t)
    if (z <= (-7.4d+213)) then
        tmp = 1.0d0 * x
    else if (z <= (-8d-40)) then
        tmp = t_1
    else if (z <= (-4d-141)) then
        tmp = x * ((y - z) / t)
    else if (z <= 1.05d+52) then
        tmp = (x / (t - 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 * x) / (z - t);
	double tmp;
	if (z <= -7.4e+213) {
		tmp = 1.0 * x;
	} else if (z <= -8e-40) {
		tmp = t_1;
	} else if (z <= -4e-141) {
		tmp = x * ((y - z) / t);
	} else if (z <= 1.05e+52) {
		tmp = (x / (t - z)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * x) / (z - t)
	tmp = 0
	if z <= -7.4e+213:
		tmp = 1.0 * x
	elif z <= -8e-40:
		tmp = t_1
	elif z <= -4e-141:
		tmp = x * ((y - z) / t)
	elif z <= 1.05e+52:
		tmp = (x / (t - z)) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * x) / Float64(z - t))
	tmp = 0.0
	if (z <= -7.4e+213)
		tmp = Float64(1.0 * x);
	elseif (z <= -8e-40)
		tmp = t_1;
	elseif (z <= -4e-141)
		tmp = Float64(x * Float64(Float64(y - z) / t));
	elseif (z <= 1.05e+52)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * x) / (z - t);
	tmp = 0.0;
	if (z <= -7.4e+213)
		tmp = 1.0 * x;
	elseif (z <= -8e-40)
		tmp = t_1;
	elseif (z <= -4e-141)
		tmp = x * ((y - z) / t);
	elseif (z <= 1.05e+52)
		tmp = (x / (t - z)) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.4e+213], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, -8e-40], t$95$1, If[LessEqual[z, -4e-141], N[(x * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+52], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.39999999999999985e213

   1. Initial program 42.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. 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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 94.0%

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

   if -7.39999999999999985e213 < z < -7.9999999999999994e-40 or 1.05e52 < z

   1. Initial program 85.9%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 69.7

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

   7. Applied rewrites 69.7%

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

   if -7.9999999999999994e-40 < z < -4.0000000000000002e-141

   1. Initial program 99.7%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 86.3

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

   5. Applied rewrites 86.3%

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

   7. Applied rewrites 86.2%

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

   if -4.0000000000000002e-141 < z < 1.05e52

   1. Initial program 92.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 78.5

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

   5. Applied rewrites 78.5%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+213}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-40}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot x}{z - t}\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+213}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{elif}\;z \leq 0.000102:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z x) (- z t))))
   (if (<= z -7.4e+213)
     (* 1.0 x)
     (if (<= z -4.8e-60)
       t_1
       (if (<= z 2.2e-131)
         (/ (* y x) t)
         (if (<= z 0.000102) (* (- y) (/ x z)) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * x) / (z - t);
	double tmp;
	if (z <= -7.4e+213) {
		tmp = 1.0 * x;
	} else if (z <= -4.8e-60) {
		tmp = t_1;
	} else if (z <= 2.2e-131) {
		tmp = (y * x) / t;
	} else if (z <= 0.000102) {
		tmp = -y * (x / 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 * x) / (z - t)
    if (z <= (-7.4d+213)) then
        tmp = 1.0d0 * x
    else if (z <= (-4.8d-60)) then
        tmp = t_1
    else if (z <= 2.2d-131) then
        tmp = (y * x) / t
    else if (z <= 0.000102d0) then
        tmp = -y * (x / 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 * x) / (z - t);
	double tmp;
	if (z <= -7.4e+213) {
		tmp = 1.0 * x;
	} else if (z <= -4.8e-60) {
		tmp = t_1;
	} else if (z <= 2.2e-131) {
		tmp = (y * x) / t;
	} else if (z <= 0.000102) {
		tmp = -y * (x / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * x) / (z - t)
	tmp = 0
	if z <= -7.4e+213:
		tmp = 1.0 * x
	elif z <= -4.8e-60:
		tmp = t_1
	elif z <= 2.2e-131:
		tmp = (y * x) / t
	elif z <= 0.000102:
		tmp = -y * (x / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * x) / Float64(z - t))
	tmp = 0.0
	if (z <= -7.4e+213)
		tmp = Float64(1.0 * x);
	elseif (z <= -4.8e-60)
		tmp = t_1;
	elseif (z <= 2.2e-131)
		tmp = Float64(Float64(y * x) / t);
	elseif (z <= 0.000102)
		tmp = Float64(Float64(-y) * Float64(x / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * x) / (z - t);
	tmp = 0.0;
	if (z <= -7.4e+213)
		tmp = 1.0 * x;
	elseif (z <= -4.8e-60)
		tmp = t_1;
	elseif (z <= 2.2e-131)
		tmp = (y * x) / t;
	elseif (z <= 0.000102)
		tmp = -y * (x / z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.4e+213], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, -4.8e-60], t$95$1, If[LessEqual[z, 2.2e-131], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 0.000102], N[((-y) * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.39999999999999985e213

   1. Initial program 42.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. 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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 94.0%

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

   if -7.39999999999999985e213 < z < -4.80000000000000019e-60 or 1.01999999999999999e-4 < z

   1. Initial program 87.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 67.1

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

   7. Applied rewrites 67.1%

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

   if -4.80000000000000019e-60 < z < 2.2e-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 2.2e-131 < z < 1.01999999999999999e-4

   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. associate-/l* N/A

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

      16. *-commutative N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 59.6%

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

   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 4 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+213}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{elif}\;z \leq 0.000102:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+213}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-40} \lor \neg \left(z \leq 2.35 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7.4e+213)
   (* 1.0 x)
   (if (or (<= z -8e-40) (not (<= z 2.35e+24)))
     (/ (* z x) (- z t))
     (* x (/ (- y z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.4e+213) {
		tmp = 1.0 * x;
	} else if ((z <= -8e-40) || !(z <= 2.35e+24)) {
		tmp = (z * x) / (z - t);
	} else {
		tmp = x * ((y - z) / t);
	}
	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 <= (-7.4d+213)) then
        tmp = 1.0d0 * x
    else if ((z <= (-8d-40)) .or. (.not. (z <= 2.35d+24))) then
        tmp = (z * x) / (z - t)
    else
        tmp = x * ((y - z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.4e+213) {
		tmp = 1.0 * x;
	} else if ((z <= -8e-40) || !(z <= 2.35e+24)) {
		tmp = (z * x) / (z - t);
	} else {
		tmp = x * ((y - z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7.4e+213:
		tmp = 1.0 * x
	elif (z <= -8e-40) or not (z <= 2.35e+24):
		tmp = (z * x) / (z - t)
	else:
		tmp = x * ((y - z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7.4e+213)
		tmp = Float64(1.0 * x);
	elseif ((z <= -8e-40) || !(z <= 2.35e+24))
		tmp = Float64(Float64(z * x) / Float64(z - t));
	else
		tmp = Float64(x * Float64(Float64(y - z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7.4e+213)
		tmp = 1.0 * x;
	elseif ((z <= -8e-40) || ~((z <= 2.35e+24)))
		tmp = (z * x) / (z - t);
	else
		tmp = x * ((y - z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7.4e+213], N[(1.0 * x), $MachinePrecision], If[Or[LessEqual[z, -8e-40], N[Not[LessEqual[z, 2.35e+24]], $MachinePrecision]], N[(N[(z * x), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.39999999999999985e213

   1. Initial program 42.0%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. 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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 94.0%

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

   if -7.39999999999999985e213 < z < -7.9999999999999994e-40 or 2.35e24 < z

   1. Initial program 86.7%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. 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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 68.6

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

   7. Applied rewrites 68.6%

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

   if -7.9999999999999994e-40 < z < 2.35e24

   1. Initial program 93.7%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 68.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+213}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-40} \lor \neg \left(z \leq 2.35 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{z \cdot x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-33}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+24}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.5e-33)
   (* 1.0 x)
   (if (<= z 2.2e-131)
     (/ (* y x) t)
     (if (<= z 1.8e+24) (* (- y) (/ x z)) (* 1.0 x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.5e-33) {
		tmp = 1.0 * x;
	} else if (z <= 2.2e-131) {
		tmp = (y * x) / t;
	} else if (z <= 1.8e+24) {
		tmp = -y * (x / z);
	} 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-33)) then
        tmp = 1.0d0 * x
    else if (z <= 2.2d-131) then
        tmp = (y * x) / t
    else if (z <= 1.8d+24) then
        tmp = -y * (x / z)
    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-33) {
		tmp = 1.0 * x;
	} else if (z <= 2.2e-131) {
		tmp = (y * x) / t;
	} else if (z <= 1.8e+24) {
		tmp = -y * (x / z);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.5e-33:
		tmp = 1.0 * x
	elif z <= 2.2e-131:
		tmp = (y * x) / t
	elif z <= 1.8e+24:
		tmp = -y * (x / z)
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.5e-33)
		tmp = Float64(1.0 * x);
	elseif (z <= 2.2e-131)
		tmp = Float64(Float64(y * x) / t);
	elseif (z <= 1.8e+24)
		tmp = Float64(Float64(-y) * Float64(x / z));
	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-33)
		tmp = 1.0 * x;
	elseif (z <= 2.2e-131)
		tmp = (y * x) / t;
	elseif (z <= 1.8e+24)
		tmp = -y * (x / z);
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.5e-33], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 2.2e-131], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.8e+24], N[((-y) * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5000000000000001e-33 or 1.79999999999999992e24 < z

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 61.5%

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

   if -1.5000000000000001e-33 < z < 2.2e-131

   1. Initial program 93.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 66.2

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

   5. Applied rewrites 66.2%

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

   if 2.2e-131 < z < 1.79999999999999992e24

   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. associate-/l* N/A

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

      16. *-commutative N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 60.5%

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

   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%

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

Alternative 11: 89.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.35e+136) (not (<= z 4.9e+182)))
   (* (/ z (- z t)) x)
   (* (/ x (- t z)) (- y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.35e+136) || !(z <= 4.9e+182)) {
		tmp = (z / (z - t)) * x;
	} else {
		tmp = (x / (t - z)) * (y - z);
	}
	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 <= (-4.35d+136)) .or. (.not. (z <= 4.9d+182))) then
        tmp = (z / (z - t)) * x
    else
        tmp = (x / (t - z)) * (y - z)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.35e+136) or not (z <= 4.9e+182):
		tmp = (z / (z - t)) * x
	else:
		tmp = (x / (t - z)) * (y - z)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.35e+136], N[Not[LessEqual[z, 4.9e+182]], $MachinePrecision]], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]

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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 95.3

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

   9. 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. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.35 \cdot 10^{+136} \lor \neg \left(z \leq 4.9 \cdot 10^{+182}\right):\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-33} \lor \neg \left(z \leq 5.2 \cdot 10^{+70}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.5e-33) (not (<= z 5.2e+70))) (* 1.0 x) (/ (* y x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.5e-33) || !(z <= 5.2e+70)) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * x) / t;
	}
	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-33)) .or. (.not. (z <= 5.2d+70))) then
        tmp = 1.0d0 * x
    else
        tmp = (y * x) / t
    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-33) || !(z <= 5.2e+70)) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * x) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.5e-33) or not (z <= 5.2e+70):
		tmp = 1.0 * x
	else:
		tmp = (y * x) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.5e-33) || !(z <= 5.2e+70))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(y * x) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.5e-33) || ~((z <= 5.2e+70)))
		tmp = 1.0 * x;
	else
		tmp = (y * x) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.5e-33], N[Not[LessEqual[z, 5.2e+70]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5000000000000001e-33 or 5.2000000000000001e70 < 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. 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 63.7%

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

   if -1.5000000000000001e-33 < z < 5.2000000000000001e70

   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. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-33} \lor \neg \left(z \leq 5.2 \cdot 10^{+70}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-39} \lor \neg \left(z \leq 5.2 \cdot 10^{+70}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.05e-39) (not (<= z 5.2e+70))) (* 1.0 x) (* (/ x t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.05e-39) || !(z <= 5.2e+70)) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / t) * y;
	}
	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 <= (-1.05d-39)) .or. (.not. (z <= 5.2d+70))) then
        tmp = 1.0d0 * x
    else
        tmp = (x / t) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.05e-39) || !(z <= 5.2e+70)) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / t) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.05e-39) or not (z <= 5.2e+70):
		tmp = 1.0 * x
	else:
		tmp = (x / t) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.05e-39) || !(z <= 5.2e+70))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(x / t) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.05e-39) || ~((z <= 5.2e+70)))
		tmp = 1.0 * x;
	else
		tmp = (x / t) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.05e-39], N[Not[LessEqual[z, 5.2e+70]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(x / t), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.04999999999999997e-39 or 5.2000000000000001e70 < 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. 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 62.7%

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

   if -1.04999999999999997e-39 < z < 5.2000000000000001e70

   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. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-39} \lor \neg \left(z \leq 5.2 \cdot 10^{+70}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \end{array} \]
5. 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. 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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lower-*.f64 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. neg-sub0 N/A

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

   10. lift--.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. associate--r+ N/A

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

   14. neg-sub0 N/A

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

   15. remove-double-neg N/A

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

   16. lower--.f64 N/A

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

   17. neg-sub0 N/A

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

   18. lift--.f64 N/A

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

   19. sub-neg N/A

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

   20. +-commutative N/A

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

   21. associate--r+ N/A

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

   22. neg-sub0 N/A

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

   23. remove-double-neg N/A

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

   24. lower--.f64 96.8

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

6. Applied rewrites 96.8%

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

7. Taylor expanded in z around inf

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

9. Applied rewrites 33.9%

   \[\leadsto \color{blue}{1} \cdot x \]
10. 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)))