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

Percentage Accurate: 88.0% → 99.9%

Time: 9.1s

Alternatives: 10

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

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

FPCore

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

C

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

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 + (-1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

2. Taylor expanded in x around inf 95.9%

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

   1. +-commutative 95.9%

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

   2. distribute-lft-in 95.9%

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

   3. *-commutative 95.9%

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

   4. *-commutative 95.9%

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

   5. associate-*r* 95.9%

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

   6. neg-mul-1 95.9%

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

   7. *-rgt-identity 95.9%

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

   8. remove-double-neg 95.9%

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

   9. neg-mul-1 95.9%

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

   10. distribute-rgt-in 95.9%

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

   11. neg-mul-1 95.9%

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

   12. metadata-eval 95.9%

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

   13. sub-neg 95.9%

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

   14. associate-*r* 95.9%

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

   15. associate-*r/ 95.9%

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

   16. mul-1-neg 95.9%

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

   17. unsub-neg 95.9%

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

   18. associate-/l* 96.5%

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

   19. associate-/r/ 99.9%

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

   20. sub-neg 99.9%

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

   21. metadata-eval 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 80.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ t_1 := x \cdot \frac{-y}{z}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))) (t_1 (* x (/ (- y) z))))
   (if (<= y -6.8e+66)
     t_0
     (if (<= y -1.85e+21)
       t_1
       (if (<= y 7.6e+18) t_0 (if (<= y 2.15e+64) t_1 (- y (/ x z))))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double t_1 = x * (-y / z);
	double tmp;
	if (y <= -6.8e+66) {
		tmp = t_0;
	} else if (y <= -1.85e+21) {
		tmp = t_1;
	} else if (y <= 7.6e+18) {
		tmp = t_0;
	} else if (y <= 2.15e+64) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y + (x / z)
    t_1 = x * (-y / z)
    if (y <= (-6.8d+66)) then
        tmp = t_0
    else if (y <= (-1.85d+21)) then
        tmp = t_1
    else if (y <= 7.6d+18) then
        tmp = t_0
    else if (y <= 2.15d+64) then
        tmp = t_1
    else
        tmp = y - (x / 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 t_1 = x * (-y / z);
	double tmp;
	if (y <= -6.8e+66) {
		tmp = t_0;
	} else if (y <= -1.85e+21) {
		tmp = t_1;
	} else if (y <= 7.6e+18) {
		tmp = t_0;
	} else if (y <= 2.15e+64) {
		tmp = t_1;
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x / z)
	t_1 = x * (-y / z)
	tmp = 0
	if y <= -6.8e+66:
		tmp = t_0
	elif y <= -1.85e+21:
		tmp = t_1
	elif y <= 7.6e+18:
		tmp = t_0
	elif y <= 2.15e+64:
		tmp = t_1
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x / z))
	t_1 = Float64(x * Float64(Float64(-y) / z))
	tmp = 0.0
	if (y <= -6.8e+66)
		tmp = t_0;
	elseif (y <= -1.85e+21)
		tmp = t_1;
	elseif (y <= 7.6e+18)
		tmp = t_0;
	elseif (y <= 2.15e+64)
		tmp = t_1;
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (x / z);
	t_1 = x * (-y / z);
	tmp = 0.0;
	if (y <= -6.8e+66)
		tmp = t_0;
	elseif (y <= -1.85e+21)
		tmp = t_1;
	elseif (y <= 7.6e+18)
		tmp = t_0;
	elseif (y <= 2.15e+64)
		tmp = t_1;
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.8e+66], t$95$0, If[LessEqual[y, -1.85e+21], t$95$1, If[LessEqual[y, 7.6e+18], t$95$0, If[LessEqual[y, 2.15e+64], t$95$1, N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.8000000000000006e66 or -1.85e21 < y < 7.6e18

   1. Initial program 92.7%

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

   2. Taylor expanded in z around inf 85.5%

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

   3. Taylor expanded in x around 0 91.7%

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

   if -6.8000000000000006e66 < y < -1.85e21 or 7.6e18 < y < 2.1499999999999999e64

   1. Initial program 91.8%

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

   2. Taylor expanded in y around inf 91.8%

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

      1. associate-/l* 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in z around 0 75.6%

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

      1. associate-*r/ 83.7%

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

      2. mul-1-neg 83.7%

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

      3. distribute-rgt-neg-in 83.7%

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

      4. mul-1-neg 83.7%

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

      5. associate-*r/ 83.7%

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

      6. neg-mul-1 83.7%

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

   7. Simplified 83.7%

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

   if 2.1499999999999999e64 < y

   1. Initial program 76.9%

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

   2. Taylor expanded in z around inf 38.6%

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

   3. Taylor expanded in x around 0 57.1%

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

      1. frac-2neg 57.1%

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

      2. div-inv 57.1%

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

      3. add-sqr-sqrt 29.0%

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

      4. sqrt-unprod 63.4%

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

      5. sqr-neg 63.4%

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

      6. sqrt-unprod 36.5%

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

      7. add-sqr-sqrt 69.8%

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

      8. cancel-sign-sub-inv 69.8%

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

      9. div-inv 69.8%

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

   5. Applied egg-rr 69.8%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+66}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+18}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \]

Alternative 3: 95.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(Float64(z - x) * Float64(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 ((y <= -1.0) || ~((y <= 1.0)))
		tmp = (z - x) * (y / 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, 1.0]], $MachinePrecision]], N[(N[(z - x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 77.9%

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

   2. Taylor expanded in y around inf 76.7%

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

      1. associate-/l* 98.6%

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

      2. associate-/r/ 90.3%

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

   4. Simplified 90.3%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 99.0%

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

   3. Taylor expanded in x around 0 99.0%

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

4. Final simplification 94.9%

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

Alternative 4: 95.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y - (x / (z / y));
	} 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 <= 1.0d0))) then
        tmp = y - (x / (z / y))
    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 <= 1.0)) {
		tmp = y - (x / (z / y));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(y - Float64(x / Float64(z / y)));
	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 <= 1.0)))
		tmp = y - (x / (z / y));
	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, 1.0]], $MachinePrecision]], N[(y - N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 77.9%

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

   2. Taylor expanded in x around inf 91.2%

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

      1. +-commutative 91.2%

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

      2. distribute-lft-in 91.2%

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

      3. *-commutative 91.2%

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

      4. *-commutative 91.2%

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

      5. associate-*r* 91.2%

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

      6. neg-mul-1 91.2%

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

      7. *-rgt-identity 91.2%

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

      8. remove-double-neg 91.2%

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

      9. neg-mul-1 91.2%

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

      10. distribute-rgt-in 91.2%

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

      11. neg-mul-1 91.2%

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

      12. metadata-eval 91.2%

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

      13. sub-neg 91.2%

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

      14. associate-*r* 91.2%

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

      15. associate-*r/ 91.2%

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

      16. mul-1-neg 91.2%

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

      17. unsub-neg 91.2%

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

      18. associate-/l* 92.5%

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

      19. associate-/r/ 99.9%

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

      20. sub-neg 99.9%

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

      21. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around inf 90.0%

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

      1. associate-/l* 91.3%

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

   7. Simplified 91.3%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 99.0%

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

   3. Taylor expanded in x around 0 99.0%

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

4. Final simplification 95.4%

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

Alternative 5: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		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 <= 1.0d0))) 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 <= 1.0)) {
		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 <= 1.0):
		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 <= 1.0))
		tmp = Float64(y / Float64(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 <= 1.0)))
		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, 1.0]], $MachinePrecision]], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 77.9%

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

   2. Taylor expanded in y around inf 76.7%

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

      1. associate-/l* 98.6%

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

   4. Simplified 98.6%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 99.0%

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

   3. Taylor expanded in x around 0 99.0%

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

4. Final simplification 98.9%

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

Alternative 6: 95.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.0) {
		tmp = y - (x * (y / z));
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = (z - x) * (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) :: tmp
    if (y <= (-1.0d0)) then
        tmp = y - (x * (y / z))
    else if (y <= 1.0d0) then
        tmp = y + (x / z)
    else
        tmp = (z - x) * (y / z)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 76.8%

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

   2. Taylor expanded in x around inf 96.5%

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

      1. +-commutative 96.5%

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

      2. distribute-lft-in 96.5%

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

      3. *-commutative 96.5%

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

      4. *-commutative 96.5%

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

      5. associate-*r* 96.5%

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

      6. neg-mul-1 96.5%

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

      7. *-rgt-identity 96.5%

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

      8. remove-double-neg 96.5%

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

      9. neg-mul-1 96.5%

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

      10. distribute-rgt-in 96.5%

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

      11. neg-mul-1 96.5%

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

      12. metadata-eval 96.5%

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

      13. sub-neg 96.5%

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

      14. associate-*r* 96.5%

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

      15. associate-*r/ 96.5%

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

      16. mul-1-neg 96.5%

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

      17. unsub-neg 96.5%

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

      18. associate-/l* 95.1%

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

      19. associate-/r/ 99.9%

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

      20. sub-neg 99.9%

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

      21. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. associate-*l/ 96.5%

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

      2. clear-num 96.5%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in y around inf 95.0%

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

      1. associate-*r/ 92.0%

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

   9. Simplified 92.0%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 99.0%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 1 < y

   1. Initial program 78.9%

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

   2. Taylor expanded in y around inf 78.0%

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

      1. associate-/l* 98.9%

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

      2. associate-/r/ 88.7%

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

   4. Simplified 88.7%

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

4. Final simplification 94.9%

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

Alternative 7: 60.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-51}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.4e-51) y (if (<= y 7e-57) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.4e-51) {
		tmp = y;
	} else if (y <= 7e-57) {
		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 (y <= (-5.4d-51)) then
        tmp = y
    else if (y <= 7d-57) 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 (y <= -5.4e-51) {
		tmp = y;
	} else if (y <= 7e-57) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.4e-51:
		tmp = y
	elif y <= 7e-57:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.4e-51)
		tmp = y;
	elseif (y <= 7e-57)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.4e-51)
		tmp = y;
	elseif (y <= 7e-57)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.4e-51], y, If[LessEqual[y, 7e-57], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -5.3999999999999994e-51 or 6.99999999999999983e-57 < y

   1. Initial program 81.7%

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

   2. Taylor expanded in x around 0 56.9%

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

   if -5.3999999999999994e-51 < y < 6.99999999999999983e-57

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 76.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-51}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8: 81.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(y + Float64(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)
		tmp = y + (x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 93.1%

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

   2. Taylor expanded in z around inf 82.8%

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

   3. Taylor expanded in x around 0 88.7%

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

   if 1 < y

   1. Initial program 78.9%

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

   2. Taylor expanded in z around inf 35.1%

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

   3. Taylor expanded in x around 0 49.5%

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

      1. frac-2neg 49.5%

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

      2. div-inv 49.5%

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

      3. add-sqr-sqrt 24.4%

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

      4. sqrt-unprod 54.9%

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

      5. sqr-neg 54.9%

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

      6. sqrt-unprod 32.4%

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

      7. add-sqr-sqrt 62.0%

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

      8. cancel-sign-sub-inv 62.0%

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

      9. div-inv 62.0%

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

   5. Applied egg-rr 62.0%

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

4. Final simplification 82.2%

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

Alternative 9: 78.0% 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 89.6%

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

2. Taylor expanded in z around inf 71.1%

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

3. Taylor expanded in x around 0 79.1%

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

4. Final simplification 79.1%

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

Alternative 10: 41.0% 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 89.6%

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

2. Taylor expanded in x around 0 43.1%

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

3. Final simplification 43.1%

   \[\leadsto y \]

Developer target: 94.0% 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 2023336 
(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))