Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Percentage Accurate: 88.0% → 99.9%

Time: 5.6s

Alternatives: 9

Speedup: 1.0×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / 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 - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / 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 - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in x around -inf 97.7%

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

   1. mul-1-neg 97.7%

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

   2. unsub-neg 97.7%

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

   3. associate-/l* 95.1%

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

   4. associate-/r/ 99.9%

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

   5. sub-neg 99.9%

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

   6. metadata-eval 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-10} \lor \neg \left(z \leq 1.7 \cdot 10^{+54}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))))
   (if (<= z -2.7e-40)
     t_0
     (if (<= z 9e-62)
       (* (/ x z) (- 1.0 y))
       (if (or (<= z 6.5e-10) (not (<= z 1.7e+54)))
         t_0
         (* x (/ (- 1.0 y) z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (z <= -2.7e-40) {
		tmp = t_0;
	} else if (z <= 9e-62) {
		tmp = (x / z) * (1.0 - y);
	} else if ((z <= 6.5e-10) || !(z <= 1.7e+54)) {
		tmp = t_0;
	} else {
		tmp = x * ((1.0 - y) / 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) :: t_0
    real(8) :: tmp
    t_0 = y + (x / z)
    if (z <= (-2.7d-40)) then
        tmp = t_0
    else if (z <= 9d-62) then
        tmp = (x / z) * (1.0d0 - y)
    else if ((z <= 6.5d-10) .or. (.not. (z <= 1.7d+54))) then
        tmp = t_0
    else
        tmp = x * ((1.0d0 - y) / z)
    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 (z <= -2.7e-40) {
		tmp = t_0;
	} else if (z <= 9e-62) {
		tmp = (x / z) * (1.0 - y);
	} else if ((z <= 6.5e-10) || !(z <= 1.7e+54)) {
		tmp = t_0;
	} else {
		tmp = x * ((1.0 - y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x / z)
	tmp = 0
	if z <= -2.7e-40:
		tmp = t_0
	elif z <= 9e-62:
		tmp = (x / z) * (1.0 - y)
	elif (z <= 6.5e-10) or not (z <= 1.7e+54):
		tmp = t_0
	else:
		tmp = x * ((1.0 - y) / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x / z))
	tmp = 0.0
	if (z <= -2.7e-40)
		tmp = t_0;
	elseif (z <= 9e-62)
		tmp = Float64(Float64(x / z) * Float64(1.0 - y));
	elseif ((z <= 6.5e-10) || !(z <= 1.7e+54))
		tmp = t_0;
	else
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (x / z);
	tmp = 0.0;
	if (z <= -2.7e-40)
		tmp = t_0;
	elseif (z <= 9e-62)
		tmp = (x / z) * (1.0 - y);
	elseif ((z <= 6.5e-10) || ~((z <= 1.7e+54)))
		tmp = t_0;
	else
		tmp = x * ((1.0 - y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.7e-40], t$95$0, If[LessEqual[z, 9e-62], N[(N[(x / z), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 6.5e-10], N[Not[LessEqual[z, 1.7e+54]], $MachinePrecision]], t$95$0, N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7e-40 or 9.00000000000000036e-62 < z < 6.5000000000000003e-10 or 1.7e54 < z

   1. Initial program 81.7%

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

   3. Taylor expanded in x around -inf 95.8%

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

      1. mul-1-neg 95.8%

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

      2. unsub-neg 95.8%

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

      3. associate-/l* 99.2%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 91.3%

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

      1. mul-1-neg 91.3%

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

      2. distribute-frac-neg 91.3%

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

   8. Simplified 91.3%

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

   if -2.7e-40 < z < 9.00000000000000036e-62

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 93.1%

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

      1. associate-/l* 88.2%

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

      2. associate-/r/ 93.1%

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

      3. mul-1-neg 93.1%

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

      4. unsub-neg 93.1%

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

   5. Simplified 93.1%

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

   if 6.5000000000000003e-10 < z < 1.7e54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-lft-neg-in 100.0%

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

      9. distribute-rgt-neg-out 100.0%

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

      10. associate-*l/ 100.0%

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

      11. associate-*r/ 100.0%

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

      12. distribute-rgt-in 100.0%

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

      13. +-commutative 100.0%

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

      14. associate-*l/ 100.0%

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

      15. distribute-lft-neg-in 100.0%

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

      16. distribute-rgt-neg-in 100.0%

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

      17. associate-*r/ 100.0%

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

      18. distribute-neg-in 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

      21. unsub-neg 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{1 - y}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-40}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-10} \lor \neg \left(z \leq 1.7 \cdot 10^{+54}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+50} \lor \neg \left(x \leq 8.5 \cdot 10^{+24}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.8e+50) (not (<= x 8.5e+24)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+50) || !(x <= 8.5e+24)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / 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 <= (-6.8d+50)) .or. (.not. (x <= 8.5d+24))) then
        tmp = x * ((1.0d0 - y) / z)
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+50) || !(x <= 8.5e+24)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.8e+50) or not (x <= 8.5e+24):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.8e+50) || !(x <= 8.5e+24))
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6.8e+50) || ~((x <= 8.5e+24)))
		tmp = x * ((1.0 - y) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.8e+50], N[Not[LessEqual[x, 8.5e+24]], $MachinePrecision]], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.7999999999999997e50 or 8.49999999999999959e24 < x

   1. Initial program 94.6%

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

   3. Taylor expanded in x around -inf 94.7%

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

      1. mul-1-neg 94.7%

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

      2. unsub-neg 94.7%

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

      3. associate-/l* 100.0%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 92.6%

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

      1. sub-neg 92.6%

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

      2. distribute-rgt-in 86.1%

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

      3. associate-*l/ 86.2%

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

      4. *-lft-identity 86.2%

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

      5. remove-double-neg 86.2%

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

      6. distribute-frac-neg 86.2%

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

      7. neg-mul-1 86.2%

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

      8. distribute-lft-neg-in 86.2%

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

      9. distribute-rgt-neg-out 86.2%

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

      10. associate-*l/ 84.4%

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

      11. associate-*r/ 76.8%

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

      12. distribute-rgt-in 92.8%

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

      13. +-commutative 92.8%

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

      14. associate-*l/ 91.0%

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

      15. distribute-lft-neg-in 91.0%

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

      16. distribute-rgt-neg-in 91.0%

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

      17. associate-*r/ 92.6%

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

      18. distribute-neg-in 92.6%

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

      19. metadata-eval 92.6%

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

      20. +-commutative 92.6%

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

      21. unsub-neg 92.6%

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

   8. Simplified 92.6%

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

   if -6.7999999999999997e50 < x < 8.49999999999999959e24

   1. Initial program 87.3%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-/l* 91.7%

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

      4. associate-/r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. distribute-frac-neg 85.5%

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

   8. Simplified 85.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+50} \lor \neg \left(x \leq 8.5 \cdot 10^{+24}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.0152\right):\\ \;\;\;\;\frac{y}{z} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.0152))) (* (/ y z) (- z x)) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.0152)) {
		tmp = (y / z) * (z - x);
	} else {
		tmp = y + (x / 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 ((y <= (-1.0d0)) .or. (.not. (y <= 0.0152d0))) then
        tmp = (y / z) * (z - x)
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.0152)) {
		tmp = (y / z) * (z - x);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.0152):
		tmp = (y / z) * (z - x)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.0152))
		tmp = Float64(Float64(y / z) * Float64(z - x));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.0152)))
		tmp = (y / z) * (z - x);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.0152]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.0152 < y

   1. Initial program 78.8%

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

   3. Taylor expanded in y around inf 77.4%

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

      1. associate-/l* 98.4%

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

      2. associate-/r/ 86.5%

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

   5. Simplified 86.5%

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

   if -1 < y < 0.0152

   1. Initial program 99.9%

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

   3. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. distribute-frac-neg 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.0152\right):\\ \;\;\;\;\frac{y}{z} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.00012\right):\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.00012))) (- y (/ (* y x) z)) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.00012)) {
		tmp = y - ((y * x) / z);
	} else {
		tmp = y + (x / 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 ((y <= (-1.0d0)) .or. (.not. (y <= 0.00012d0))) then
        tmp = y - ((y * x) / z)
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.00012)) {
		tmp = y - ((y * x) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.00012):
		tmp = y - ((y * x) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.00012))
		tmp = Float64(y - Float64(Float64(y * x) / z));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.00012)))
		tmp = y - ((y * x) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.00012]], $MachinePrecision]], N[(y - N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.20000000000000003e-4 < y

   1. Initial program 79.3%

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

   3. Taylor expanded in x around -inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. unsub-neg 95.2%

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

      3. associate-/l* 89.5%

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

      4. associate-/r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around inf 93.8%

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

   if -1 < y < 1.20000000000000003e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. distribute-frac-neg 98.4%

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

   8. Simplified 98.4%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.00012\right):\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.3e+69) (+ y (/ x z)) (/ (* y (- x)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3e+69) {
		tmp = y + (x / z);
	} else {
		tmp = (y * -x) / 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 (y <= 3.3d+69) then
        tmp = y + (x / z)
    else
        tmp = (y * -x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3e+69) {
		tmp = y + (x / z);
	} else {
		tmp = (y * -x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.3e+69:
		tmp = y + (x / z)
	else:
		tmp = (y * -x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.3e+69)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(Float64(y * Float64(-x)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.3e+69)
		tmp = y + (x / z);
	else
		tmp = (y * -x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.3e+69], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * (-x)), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2999999999999999e69

   1. Initial program 93.3%

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

   3. Taylor expanded in x around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. associate-/l* 96.5%

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

      4. associate-/r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 86.3%

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

      1. mul-1-neg 86.3%

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

      2. distribute-frac-neg 86.3%

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

   8. Simplified 86.3%

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

   if 3.2999999999999999e69 < y

   1. Initial program 75.6%

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

   3. Taylor expanded in y around inf 75.6%

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

   4. Taylor expanded in z around 0 56.4%

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

      1. mul-1-neg 56.4%

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

      2. distribute-lft-neg-out 56.4%

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

      3. *-commutative 56.4%

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

   6. Simplified 56.4%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-7}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+119}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.35e-7) y (if (<= z 1.85e+119) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e-7) {
		tmp = y;
	} else if (z <= 1.85e+119) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	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-7)) then
        tmp = y
    else if (z <= 1.85d+119) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e-7) {
		tmp = y;
	} else if (z <= 1.85e+119) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.35e-7:
		tmp = y
	elif z <= 1.85e+119:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.35e-7)
		tmp = y;
	elseif (z <= 1.85e+119)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.35e-7)
		tmp = y;
	elseif (z <= 1.85e+119)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.35e-7], y, If[LessEqual[z, 1.85e+119], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000004e-7 or 1.85e119 < z

   1. Initial program 76.7%

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

   3. Taylor expanded in x around 0 70.7%

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

   if -1.35000000000000004e-7 < z < 1.85e119

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 59.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-7}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+119}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ y (/ x z)))

C

double code(double x, double y, double z) {
	return y + (x / 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 = y + (x / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in x around -inf 97.7%

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

   1. mul-1-neg 97.7%

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

   2. unsub-neg 97.7%

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

   3. associate-/l* 95.1%

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

   4. associate-/r/ 99.9%

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

   5. sub-neg 99.9%

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

   6. metadata-eval 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in y around 0 79.8%

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

   1. mul-1-neg 79.8%

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

   2. distribute-frac-neg 79.8%

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

8. Simplified 79.8%

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

9. Final simplification 79.8%

   \[\leadsto y + \frac{x}{z} \]
10. Add Preprocessing

Alternative 9: 40.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in x around 0 37.8%

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

4. Final simplification 37.8%

   \[\leadsto y \]
5. Add Preprocessing

Developer target: 93.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y + (x / z)) - (y / (z / 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 = (y + (x / z)) - (y / (z / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024010 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

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