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

Percentage Accurate: 88.2% → 99.8%

Time: 7.0s

Alternatives: 9

Speedup: 1.0×

• 21.1% 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: 88.2% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y + \frac{1 - y}{\frac{z}{x}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return y + ((1.0 - 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 + ((1.0d0 - y) / (z / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

3. Taylor expanded in x around 0 96.6%

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

   1. +-commutative 96.6%

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

   2. mul-1-neg 96.6%

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

   3. sub-neg 96.6%

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

   4. div-sub 96.6%

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

   5. associate-*r/ 96.6%

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

   6. clear-num 96.6%

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

   7. frac-2neg 96.6%

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

   8. metadata-eval 96.6%

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

   9. associate-/r* 99.9%

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

5. Applied egg-rr 99.9%

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

   1. distribute-frac-neg2 99.9%

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

   2. distribute-neg-frac 99.9%

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

   3. metadata-eval 99.9%

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

   4. associate-/r/ 99.9%

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

   5. associate-*l/ 99.9%

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

   6. *-lft-identity 99.9%

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

7. Simplified 99.9%

   \[\leadsto y + \color{blue}{\frac{1 - y}{\frac{z}{x}}} \]
8. Add Preprocessing

Alternative 2: 62.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-17}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-98} \lor \neg \left(y \leq 7.5 \cdot 10^{-82}\right) \land y \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.55e-17)
   (* z (/ y z))
   (if (or (<= y 3e-98) (and (not (<= y 7.5e-82)) (<= y 5.8e-14))) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e-17) {
		tmp = z * (y / z);
	} else if ((y <= 3e-98) || (!(y <= 7.5e-82) && (y <= 5.8e-14))) {
		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 <= (-1.55d-17)) then
        tmp = z * (y / z)
    else if ((y <= 3d-98) .or. (.not. (y <= 7.5d-82)) .and. (y <= 5.8d-14)) 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 <= -1.55e-17) {
		tmp = z * (y / z);
	} else if ((y <= 3e-98) || (!(y <= 7.5e-82) && (y <= 5.8e-14))) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.55e-17:
		tmp = z * (y / z)
	elif (y <= 3e-98) or (not (y <= 7.5e-82) and (y <= 5.8e-14)):
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.55e-17)
		tmp = Float64(z * Float64(y / z));
	elseif ((y <= 3e-98) || (!(y <= 7.5e-82) && (y <= 5.8e-14)))
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.55e-17)
		tmp = z * (y / z);
	elseif ((y <= 3e-98) || (~((y <= 7.5e-82)) && (y <= 5.8e-14)))
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.55e-17], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3e-98], And[N[Not[LessEqual[y, 7.5e-82]], $MachinePrecision], LessEqual[y, 5.8e-14]]], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5499999999999999e-17

   1. Initial program 79.0%

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

   3. Taylor expanded in y around inf 79.0%

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

   4. Taylor expanded in z around inf 30.1%

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

      1. *-commutative 30.1%

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

   6. Simplified 30.1%

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

      1. associate-/l* 52.5%

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

      2. *-commutative 52.5%

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

   8. Applied egg-rr 52.5%

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

   if -1.5499999999999999e-17 < y < 3e-98 or 7.4999999999999997e-82 < y < 5.8000000000000005e-14

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 74.4%

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

   if 3e-98 < y < 7.4999999999999997e-82 or 5.8000000000000005e-14 < y

   1. Initial program 74.7%

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

   3. Taylor expanded in x around 0 66.5%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-17}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-98} \lor \neg \left(y \leq 7.5 \cdot 10^{-82}\right) \land y \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-17}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-99} \lor \neg \left(y \leq 8.5 \cdot 10^{-82}\right) \land y \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8e-17)
   y
   (if (or (<= y 2e-99) (and (not (<= y 8.5e-82)) (<= y 1.55e-7))) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8e-17) {
		tmp = y;
	} else if ((y <= 2e-99) || (!(y <= 8.5e-82) && (y <= 1.55e-7))) {
		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 <= (-8d-17)) then
        tmp = y
    else if ((y <= 2d-99) .or. (.not. (y <= 8.5d-82)) .and. (y <= 1.55d-7)) 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 <= -8e-17) {
		tmp = y;
	} else if ((y <= 2e-99) || (!(y <= 8.5e-82) && (y <= 1.55e-7))) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8e-17:
		tmp = y
	elif (y <= 2e-99) or (not (y <= 8.5e-82) and (y <= 1.55e-7)):
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8e-17)
		tmp = y;
	elseif ((y <= 2e-99) || (!(y <= 8.5e-82) && (y <= 1.55e-7)))
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8e-17)
		tmp = y;
	elseif ((y <= 2e-99) || (~((y <= 8.5e-82)) && (y <= 1.55e-7)))
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8e-17], y, If[Or[LessEqual[y, 2e-99], And[N[Not[LessEqual[y, 8.5e-82]], $MachinePrecision], LessEqual[y, 1.55e-7]]], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000057e-17 or 2e-99 < y < 8.4999999999999997e-82 or 1.55e-7 < y

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0 56.9%

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

   if -8.00000000000000057e-17 < y < 2e-99 or 8.4999999999999997e-82 < y < 1.55e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 74.4%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-17}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-99} \lor \neg \left(y \leq 8.5 \cdot 10^{-82}\right) \land y \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{-z}\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+80}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x (- z)))))
   (if (<= y -8.8e+86)
     t_0
     (if (<= y 2.15e+80) (+ y (/ x z)) (if (<= y 1.2e+116) t_0 y)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / -z);
	double tmp;
	if (y <= -8.8e+86) {
		tmp = t_0;
	} else if (y <= 2.15e+80) {
		tmp = y + (x / z);
	} else if (y <= 1.2e+116) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y * (x / -z)
    if (y <= (-8.8d+86)) then
        tmp = t_0
    else if (y <= 2.15d+80) then
        tmp = y + (x / z)
    else if (y <= 1.2d+116) then
        tmp = t_0
    else
        tmp = y
    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 <= -8.8e+86) {
		tmp = t_0;
	} else if (y <= 2.15e+80) {
		tmp = y + (x / z);
	} else if (y <= 1.2e+116) {
		tmp = t_0;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / -z)
	tmp = 0
	if y <= -8.8e+86:
		tmp = t_0
	elif y <= 2.15e+80:
		tmp = y + (x / z)
	elif y <= 1.2e+116:
		tmp = t_0
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / Float64(-z)))
	tmp = 0.0
	if (y <= -8.8e+86)
		tmp = t_0;
	elseif (y <= 2.15e+80)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 1.2e+116)
		tmp = t_0;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / -z);
	tmp = 0.0;
	if (y <= -8.8e+86)
		tmp = t_0;
	elseif (y <= 2.15e+80)
		tmp = y + (x / z);
	elseif (y <= 1.2e+116)
		tmp = t_0;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.8e+86], t$95$0, If[LessEqual[y, 2.15e+80], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+116], t$95$0, y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.80000000000000013e86 or 2.15000000000000002e80 < y < 1.2e116

   1. Initial program 71.3%

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

   3. Taylor expanded in x around inf 62.7%

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

      1. mul-1-neg 62.7%

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

      2. unsub-neg 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in y around inf 62.7%

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

      1. mul-1-neg 62.7%

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

      2. distribute-rgt-neg-out 62.7%

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

   8. Simplified 62.7%

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

      1. frac-2neg 62.7%

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

      2. distribute-frac-neg 62.7%

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

      3. add-sqr-sqrt 16.5%

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

      4. sqrt-unprod 17.5%

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

      5. sqr-neg 17.5%

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

      6. sqrt-unprod 0.8%

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

      7. add-sqr-sqrt 0.9%

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

      8. distribute-rgt-neg-in 0.9%

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

      9. frac-2neg 0.9%

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

      10. *-commutative 0.9%

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

      11. associate-/l* 1.0%

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

      12. add-sqr-sqrt 0.9%

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

      13. sqrt-unprod 19.1%

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

      14. sqr-neg 19.1%

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

      15. sqrt-unprod 18.3%

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

      16. add-sqr-sqrt 72.0%

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

   10. Applied egg-rr 72.0%

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

   if -8.80000000000000013e86 < y < 2.15000000000000002e80

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 99.9%

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

   4. Taylor expanded in y around 0 96.5%

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

   if 1.2e116 < y

   1. Initial program 65.5%

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

   3. Taylor expanded in x around 0 71.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+80}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+116}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 6.2 \cdot 10^{-5}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 6.2e-5))) (* y (- 1.0 (/ x z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 6.2e-5)) {
		tmp = y * (1.0 - (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 <= 6.2d-5))) then
        tmp = y * (1.0d0 - (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 <= 6.2e-5)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 6.2e-5):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 6.2e-5))
		tmp = Float64(y * Float64(1.0 - 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) || ~((y <= 6.2e-5)))
		tmp = y * (1.0 - (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, 6.2e-5]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 6.20000000000000027e-5 < y

   1. Initial program 74.4%

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

   3. Taylor expanded in y around inf 73.9%

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

      1. associate-/l* 99.4%

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

      2. div-sub 99.4%

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

      3. *-inverses 99.4%

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

   5. Simplified 99.4%

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

   if -1 < y < 6.20000000000000027e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.8%

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

   4. Taylor expanded in y around 0 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 6.2 \cdot 10^{-5}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{+142} \lor \neg \left(x \leq 1.25 \cdot 10^{+18}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.46e+142) (not (<= x 1.25e+18)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.46e+142) || !(x <= 1.25e+18)) {
		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 <= (-1.46d+142)) .or. (.not. (x <= 1.25d+18))) 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 <= -1.46e+142) || !(x <= 1.25e+18)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.46e+142) or not (x <= 1.25e+18):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.46e+142) || !(x <= 1.25e+18))
		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 <= -1.46e+142) || ~((x <= 1.25e+18)))
		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, -1.46e+142], N[Not[LessEqual[x, 1.25e+18]], $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 < -1.4600000000000001e142 or 1.25e18 < x

   1. Initial program 90.3%

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

   3. Taylor expanded in x around inf 85.5%

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

      1. associate-/l* 88.2%

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

      2. mul-1-neg 88.2%

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

      3. unsub-neg 88.2%

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

   5. Simplified 88.2%

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

   if -1.4600000000000001e142 < x < 1.25e18

   1. Initial program 86.4%

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

   3. Taylor expanded in x around 0 94.6%

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

   4. Taylor expanded in y around 0 89.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{+142} \lor \neg \left(x \leq 1.25 \cdot 10^{+18}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.0)
   (* y (- 1.0 (/ x z)))
   (if (<= y 6.2e-5) (+ y (/ x z)) (- y (* y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 6.2e-5) {
		tmp = y + (x / z);
	} else {
		tmp = y - (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 * (1.0d0 - (x / z))
    else if (y <= 6.2d-5) then
        tmp = y + (x / z)
    else
        tmp = y - (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 * (1.0 - (x / z));
	} else if (y <= 6.2e-5) {
		tmp = y + (x / z);
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.0:
		tmp = y * (1.0 - (x / z))
	elif y <= 6.2e-5:
		tmp = y + (x / z)
	else:
		tmp = y - (y * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	elseif (y <= 6.2e-5)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - 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 * (1.0 - (x / z));
	elseif (y <= 6.2e-5)
		tmp = y + (x / z);
	else
		tmp = y - (y * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.0], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e-5], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 77.4%

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

   3. Taylor expanded in y around inf 76.2%

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

      1. associate-/l* 98.7%

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

      2. div-sub 98.7%

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

      3. *-inverses 98.7%

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

   5. Simplified 98.7%

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

   if -1 < y < 6.20000000000000027e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.8%

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

   4. Taylor expanded in y around 0 99.3%

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

   if 6.20000000000000027e-5 < y

   1. Initial program 72.1%

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

   3. Taylor expanded in x around 0 91.5%

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

      1. +-commutative 91.5%

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

      2. mul-1-neg 91.5%

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

      3. sub-neg 91.5%

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

      4. div-sub 91.5%

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

      5. associate-*r/ 94.2%

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

      6. clear-num 94.2%

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

      7. frac-2neg 94.2%

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

      8. metadata-eval 94.2%

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

      9. associate-/r* 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. distribute-frac-neg2 99.9%

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

      2. distribute-neg-frac 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-/r/ 99.9%

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

      5. associate-*l/ 99.9%

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

      6. *-lft-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around inf 94.2%

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

      1. associate-*r/ 94.2%

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

      2. *-commutative 94.2%

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

      3. mul-1-neg 94.2%

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

      4. distribute-rgt-neg-out 94.2%

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

      5. associate-*r/ 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 87.9%

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

3. Taylor expanded in x around 0 96.6%

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

4. Taylor expanded in y around 0 82.0%

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

Alternative 9: 39.8% 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 87.9%

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

3. Taylor expanded in x around 0 42.8%

   \[\leadsto \color{blue}{y} \]
4. Add Preprocessing

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 2024086 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :alt
  (- (+ y (/ x z)) (/ y (/ z x)))

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