Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Percentage Accurate: 88.4% → 99.8%

Time: 8.2s

Alternatives: 9

Speedup: 0.6×

• 20.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ -1.0 (/ (+ y 1.0) z)))))
   (if (<= z -1.8e+36) t_0 (if (<= z 5e-19) (* (+ (- y z) 1.0) (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (-1.0 + ((y + 1.0) / z));
	double tmp;
	if (z <= -1.8e+36) {
		tmp = t_0;
	} else if (z <= 5e-19) {
		tmp = ((y - z) + 1.0) * (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) + ((y + 1.0d0) / z))
    if (z <= (-1.8d+36)) then
        tmp = t_0
    else if (z <= 5d-19) then
        tmp = ((y - z) + 1.0d0) * (x / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (-1.0 + ((y + 1.0) / z));
	double tmp;
	if (z <= -1.8e+36) {
		tmp = t_0;
	} else if (z <= 5e-19) {
		tmp = ((y - z) + 1.0) * (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (-1.0 + ((y + 1.0) / z))
	tmp = 0
	if z <= -1.8e+36:
		tmp = t_0
	elif z <= 5e-19:
		tmp = ((y - z) + 1.0) * (x / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-1.0 + Float64(Float64(y + 1.0) / z)))
	tmp = 0.0
	if (z <= -1.8e+36)
		tmp = t_0;
	elseif (z <= 5e-19)
		tmp = Float64(Float64(Float64(y - z) + 1.0) * Float64(x / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (-1.0 + ((y + 1.0) / z));
	tmp = 0.0;
	if (z <= -1.8e+36)
		tmp = t_0;
	elseif (z <= 5e-19)
		tmp = ((y - z) + 1.0) * (x / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(-1.0 + N[(N[(y + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8e+36], t$95$0, If[LessEqual[z, 5e-19], N[(N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7999999999999999e36 or 5.0000000000000004e-19 < z

   1. Initial program 75.6%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   if -1.7999999999999999e36 < z < 5.0000000000000004e-19

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(y - z\right) + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(y - z\right), 1\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(y, z\right), 1\right), \left(\frac{x}{z}\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(y, z\right), 1\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ -1.0 (/ (+ y 1.0) z)))))
   (if (<= z -1.9e-16) t_0 (if (<= z 5e-20) (/ (* x (+ y 1.0)) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (-1.0 + ((y + 1.0) / z));
	double tmp;
	if (z <= -1.9e-16) {
		tmp = t_0;
	} else if (z <= 5e-20) {
		tmp = (x * (y + 1.0)) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) + ((y + 1.0d0) / z))
    if (z <= (-1.9d-16)) then
        tmp = t_0
    else if (z <= 5d-20) then
        tmp = (x * (y + 1.0d0)) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (-1.0 + ((y + 1.0) / z));
	double tmp;
	if (z <= -1.9e-16) {
		tmp = t_0;
	} else if (z <= 5e-20) {
		tmp = (x * (y + 1.0)) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (-1.0 + ((y + 1.0) / z))
	tmp = 0
	if z <= -1.9e-16:
		tmp = t_0
	elif z <= 5e-20:
		tmp = (x * (y + 1.0)) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-1.0 + Float64(Float64(y + 1.0) / z)))
	tmp = 0.0
	if (z <= -1.9e-16)
		tmp = t_0;
	elseif (z <= 5e-20)
		tmp = Float64(Float64(x * Float64(y + 1.0)) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (-1.0 + ((y + 1.0) / z));
	tmp = 0.0;
	if (z <= -1.9e-16)
		tmp = t_0;
	elseif (z <= 5e-20)
		tmp = (x * (y + 1.0)) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(-1.0 + N[(N[(y + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.9e-16], t$95$0, If[LessEqual[z, 5e-20], N[(N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.90000000000000006e-16 or 4.9999999999999999e-20 < z

   1. Initial program 77.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   if -1.90000000000000006e-16 < z < 4.9999999999999999e-20

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\color{blue}{y}, 1\right)\right), z\right) \]
   4. Step-by-step derivation

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+50}:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{-242}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.35e+50)
   (- 0.0 x)
   (if (<= z 7.3e-242) (* y (/ x z)) (if (<= z 1.0) (/ x z) (- 0.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e+50) {
		tmp = 0.0 - x;
	} else if (z <= 7.3e-242) {
		tmp = y * (x / z);
	} else if (z <= 1.0) {
		tmp = x / z;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.35d+50)) then
        tmp = 0.0d0 - x
    else if (z <= 7.3d-242) then
        tmp = y * (x / z)
    else if (z <= 1.0d0) then
        tmp = x / z
    else
        tmp = 0.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e+50) {
		tmp = 0.0 - x;
	} else if (z <= 7.3e-242) {
		tmp = y * (x / z);
	} else if (z <= 1.0) {
		tmp = x / z;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.35e+50:
		tmp = 0.0 - x
	elif z <= 7.3e-242:
		tmp = y * (x / z)
	elif z <= 1.0:
		tmp = x / z
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.35e+50)
		tmp = Float64(0.0 - x);
	elseif (z <= 7.3e-242)
		tmp = Float64(y * Float64(x / z));
	elseif (z <= 1.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(0.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.35e+50)
		tmp = 0.0 - x;
	elseif (z <= 7.3e-242)
		tmp = y * (x / z);
	elseif (z <= 1.0)
		tmp = x / z;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.35e+50], N[(0.0 - x), $MachinePrecision], If[LessEqual[z, 7.3e-242], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x / z), $MachinePrecision], N[(0.0 - x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e50 or 1 < z

   1. Initial program 74.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x} \]

      3. --lowering--.f64 82.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{x}\right) \]

   7. Simplified 82.7%

      \[\leadsto \color{blue}{0 - x} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-lowering-neg.f64 82.7%

         \[\leadsto \mathsf{neg.f64}\left(x\right) \]

   9. Applied egg-rr 82.7%

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

   if -1.35e50 < z < 7.3000000000000001e-242

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(y - z\right) + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(y - z\right), 1\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(y, z\right), 1\right), \left(\frac{x}{z}\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(y, z\right), 1\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{/.f64}\left(x, z\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 64.3%

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

   if 7.3000000000000001e-242 < z < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 92.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 92.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x}{z}\right), \color{blue}{x}\right) \]

      14. /-lowering-/.f64 70.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, z\right), x\right) \]

   7. Simplified 70.0%

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

   8. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 69.5%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   10. Simplified 69.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+50}:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{-242}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+50}:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-126}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.6e+50)
   (- 0.0 x)
   (if (<= z -2.2e-126) (* x (/ y z)) (if (<= z 1.0) (/ x z) (- 0.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e+50) {
		tmp = 0.0 - x;
	} else if (z <= -2.2e-126) {
		tmp = x * (y / z);
	} else if (z <= 1.0) {
		tmp = x / z;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.6d+50)) then
        tmp = 0.0d0 - x
    else if (z <= (-2.2d-126)) then
        tmp = x * (y / z)
    else if (z <= 1.0d0) then
        tmp = x / z
    else
        tmp = 0.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e+50) {
		tmp = 0.0 - x;
	} else if (z <= -2.2e-126) {
		tmp = x * (y / z);
	} else if (z <= 1.0) {
		tmp = x / z;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.6e+50:
		tmp = 0.0 - x
	elif z <= -2.2e-126:
		tmp = x * (y / z)
	elif z <= 1.0:
		tmp = x / z
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.6e+50)
		tmp = Float64(0.0 - x);
	elseif (z <= -2.2e-126)
		tmp = Float64(x * Float64(y / z));
	elseif (z <= 1.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(0.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.6e+50)
		tmp = 0.0 - x;
	elseif (z <= -2.2e-126)
		tmp = x * (y / z);
	elseif (z <= 1.0)
		tmp = x / z;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.6e+50], N[(0.0 - x), $MachinePrecision], If[LessEqual[z, -2.2e-126], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x / z), $MachinePrecision], N[(0.0 - x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.59999999999999991e50 or 1 < z

   1. Initial program 74.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x} \]

      3. --lowering--.f64 82.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{x}\right) \]

   7. Simplified 82.7%

      \[\leadsto \color{blue}{0 - x} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-lowering-neg.f64 82.7%

         \[\leadsto \mathsf{neg.f64}\left(x\right) \]

   9. Applied egg-rr 82.7%

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

   if -1.59999999999999991e50 < z < -2.20000000000000014e-126

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 97.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 56.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 56.7%

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

   if -2.20000000000000014e-126 < z < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 90.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x}{z}\right), \color{blue}{x}\right) \]

      14. /-lowering-/.f64 64.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, z\right), x\right) \]

   7. Simplified 64.4%

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

   8. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 64.2%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   10. Simplified 64.2%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+50}:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-126}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+50}:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+40}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.8e+50)
   (- 0.0 x)
   (if (<= z 2.3e+40) (/ (* x (+ y 1.0)) z) (- 0.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e+50) {
		tmp = 0.0 - x;
	} else if (z <= 2.3e+40) {
		tmp = (x * (y + 1.0)) / z;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.8d+50)) then
        tmp = 0.0d0 - x
    else if (z <= 2.3d+40) then
        tmp = (x * (y + 1.0d0)) / z
    else
        tmp = 0.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e+50) {
		tmp = 0.0 - x;
	} else if (z <= 2.3e+40) {
		tmp = (x * (y + 1.0)) / z;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.8e+50:
		tmp = 0.0 - x
	elif z <= 2.3e+40:
		tmp = (x * (y + 1.0)) / z
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.8e+50)
		tmp = Float64(0.0 - x);
	elseif (z <= 2.3e+40)
		tmp = Float64(Float64(x * Float64(y + 1.0)) / z);
	else
		tmp = Float64(0.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.8e+50)
		tmp = 0.0 - x;
	elseif (z <= 2.3e+40)
		tmp = (x * (y + 1.0)) / z;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.8e+50], N[(0.0 - x), $MachinePrecision], If[LessEqual[z, 2.3e+40], N[(N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(0.0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7999999999999998e50 or 2.29999999999999994e40 < z

   1. Initial program 74.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x} \]

      3. --lowering--.f64 84.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{x}\right) \]

   7. Simplified 84.4%

      \[\leadsto \color{blue}{0 - x} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-lowering-neg.f64 84.4%

         \[\leadsto \mathsf{neg.f64}\left(x\right) \]

   9. Applied egg-rr 84.4%

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

   if -2.7999999999999998e50 < z < 2.29999999999999994e40

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\color{blue}{y}, 1\right)\right), z\right) \]
   4. Step-by-step derivation

   5. Simplified 94.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+50}:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+40}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= y -2.7e+41) t_0 (if (<= y 3.9e+34) (- (/ x z) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (y <= -2.7e+41) {
		tmp = t_0;
	} else if (y <= 3.9e+34) {
		tmp = (x / z) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x / z)
    if (y <= (-2.7d+41)) then
        tmp = t_0
    else if (y <= 3.9d+34) then
        tmp = (x / z) - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (y <= -2.7e+41) {
		tmp = t_0;
	} else if (y <= 3.9e+34) {
		tmp = (x / z) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if y <= -2.7e+41:
		tmp = t_0
	elif y <= 3.9e+34:
		tmp = (x / z) - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (y <= -2.7e+41)
		tmp = t_0;
	elseif (y <= 3.9e+34)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	tmp = 0.0;
	if (y <= -2.7e+41)
		tmp = t_0;
	elseif (y <= 3.9e+34)
		tmp = (x / z) - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.7e+41], t$95$0, If[LessEqual[y, 3.9e+34], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e41 or 3.90000000000000019e34 < y

   1. Initial program 89.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(y - z\right) + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(y - z\right), 1\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(y, z\right), 1\right), \left(\frac{x}{z}\right)\right) \]

      6. /-lowering-/.f64 90.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(y, z\right), 1\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 90.4%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{/.f64}\left(x, z\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.9%

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

   if -2.7e41 < y < 3.90000000000000019e34

   1. Initial program 85.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 99.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x}{z}\right), \color{blue}{x}\right) \]

      14. /-lowering-/.f64 95.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, z\right), x\right) \]

   7. Simplified 95.9%

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

Alternative 7: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{-20}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.22e-20)
   (/ (* x (+ (- y z) 1.0)) z)
   (* x (+ -1.0 (/ (+ y 1.0) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.22e-20) {
		tmp = (x * ((y - z) + 1.0)) / z;
	} else {
		tmp = x * (-1.0 + ((y + 1.0) / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.22d-20) then
        tmp = (x * ((y - z) + 1.0d0)) / z
    else
        tmp = x * ((-1.0d0) + ((y + 1.0d0) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.22e-20) {
		tmp = (x * ((y - z) + 1.0)) / z;
	} else {
		tmp = x * (-1.0 + ((y + 1.0) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.22e-20:
		tmp = (x * ((y - z) + 1.0)) / z
	else:
		tmp = x * (-1.0 + ((y + 1.0) / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.22e-20)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z);
	else
		tmp = Float64(x * Float64(-1.0 + Float64(Float64(y + 1.0) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.22e-20)
		tmp = (x * ((y - z) + 1.0)) / z;
	else
		tmp = x * (-1.0 + ((y + 1.0) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.22e-20], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(-1.0 + N[(N[(y + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.22000000000000003e-20

   1. Initial program 90.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   if 1.22000000000000003e-20 < x

   1. Initial program 74.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 99.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{-20}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0) (- 0.0 x) (if (<= z 1.0) (/ x z) (- 0.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = 0.0 - x;
	} else if (z <= 1.0) {
		tmp = x / z;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = 0.0d0 - x
    else if (z <= 1.0d0) then
        tmp = x / z
    else
        tmp = 0.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = 0.0 - x;
	} else if (z <= 1.0) {
		tmp = x / z;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = 0.0 - x
	elif z <= 1.0:
		tmp = x / z
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(0.0 - x);
	elseif (z <= 1.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(0.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = 0.0 - x;
	elseif (z <= 1.0)
		tmp = x / z;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(0.0 - x), $MachinePrecision], If[LessEqual[z, 1.0], N[(x / z), $MachinePrecision], N[(0.0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 77.0%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x} \]

      3. --lowering--.f64 79.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{x}\right) \]

   7. Simplified 79.3%

      \[\leadsto \color{blue}{0 - x} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-lowering-neg.f64 79.3%

         \[\leadsto \mathsf{neg.f64}\left(x\right) \]

   9. Applied egg-rr 79.3%

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

   if -1 < z < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

      18. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      20. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

      21. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

      22. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

      23. metadata-eval 92.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

   3. Simplified 92.3%

      \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x}{z}\right), \color{blue}{x}\right) \]

      14. /-lowering-/.f64 59.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, z\right), x\right) \]

   7. Simplified 59.7%

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

   8. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 59.4%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   10. Simplified 59.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 0 - x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- 0.0 x))

C

double code(double x, double y, double z) {
	return 0.0 - x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0 - x
end function

Java

public static double code(double x, double y, double z) {
	return 0.0 - x;
}

Python

def code(x, y, z):
	return 0.0 - x

Julia

function code(x, y, z)
	return Float64(0.0 - x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0 - x;
end

Wolfram

code[x_, y_, z_] := N[(0.0 - x), $MachinePrecision]

Derivation

1. Initial program 87.1%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
2. Step-by-step derivation

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\left(y - z\right) + 1}{z}\right)}\right) \]

   3. associate-+l- N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y - \left(z - 1\right)}{z}\right)\right) \]

   4. div-sub N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} - \color{blue}{\frac{z - 1}{z}}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right)}\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{z - 1}{z}\right)\right) + \color{blue}{\frac{y}{z}}\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} - \frac{1}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\frac{z}{z} + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)\right)\right) + \frac{y}{z}\right)\right) \]

   10. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(1 + \frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{y}{z}\right)\right) \]

   11. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(1\right)}{z}\right)\right)\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{\mathsf{neg}\left(1\right)}{z}\right) + \frac{\color{blue}{y}}{z}\right)\right) \]

   13. associate-+l- N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) - \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{z} - \frac{y}{z}\right)}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z}} - \frac{y}{z}\right)\right)\right) \]

   16. div-sub N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) - y}{\color{blue}{z}}\right)\right)\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}{z}\right)\right)\right) \]

   18. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(1 + y\right)\right)}{z}\right)\right)\right) \]

   19. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + y\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

   20. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right)\right)\right) \]

   21. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) - y\right), z\right)\right)\right) \]

   22. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), y\right), z\right)\right)\right) \]

   23. metadata-eval 96.6%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, y\right), z\right)\right)\right) \]

3. Simplified 96.6%

   \[\leadsto \color{blue}{x \cdot \left(-1 - \frac{-1 - y}{z}\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(x\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{x} \]

   3. --lowering--.f64 45.9%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{x}\right) \]

7. Simplified 45.9%

   \[\leadsto \color{blue}{0 - x} \]
8. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(x\right) \]

   2. neg-lowering-neg.f64 45.9%

      \[\leadsto \mathsf{neg.f64}\left(x\right) \]

9. Applied egg-rr 45.9%

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

10. Final simplification 45.9%

    \[\leadsto 0 - x \]
11. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ 1.0 y) (/ x z)) x)))
   (if (< x -2.71483106713436e-162)
     t_0
     (if (< x 3.874108816439546e-197)
       (* (* x (+ (- y z) 1.0)) (/ 1.0 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 + y) * (x / z)) - x
	tmp = 0
	if x < -2.71483106713436e-162:
		tmp = t_0
	elif x < 3.874108816439546e-197:
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 + y) * Float64(x / z)) - x)
	tmp = 0.0
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) * Float64(1.0 / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 + y) * (x / z)) - x;
	tmp = 0.0;
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[Less[x, -2.71483106713436e-162], t$95$0, If[Less[x, 3.874108816439546e-197], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024163 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -67870776678359/25000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (+ 1 y) (/ x z)) x) (if (< x 1937054408219773/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x))))

  (/ (* x (+ (- y z) 1.0)) z))