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

Percentage Accurate: 88.4% → 99.2%

Time: 8.9s

Alternatives: 11

Speedup: 0.5×

• 21.2% 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: 88.4% 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: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+92} \lor \neg \left(y \leq 2600000\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e+92) (not (<= y 2600000.0)))
   (* y (/ (- z x) z))
   (/ (fma y (- z x) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e+92) || !(y <= 2600000.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = fma(y, (z - x), x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e+92) || !(y <= 2600000.0))
		tmp = Float64(y * Float64(Float64(z - x) / z));
	else
		tmp = Float64(fma(y, Float64(z - x), x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1e+92], N[Not[LessEqual[y, 2600000.0]], $MachinePrecision]], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(z - x), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e92 or 2.6e6 < y

   1. Initial program 78.2%

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

   3. Taylor expanded in y around inf 78.2%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   if -1e92 < y < 2.6e6

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+92} \lor \neg \left(y \leq 2600000\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+91} \lor \neg \left(y \leq 2400000\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9.5e+91) (not (<= y 2400000.0)))
   (* y (/ (- z x) z))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.5e+91) || !(y <= 2400000.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = (x + (y * (z - 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 <= (-9.5d+91)) .or. (.not. (y <= 2400000.0d0))) then
        tmp = y * ((z - x) / z)
    else
        tmp = (x + (y * (z - x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.5e+91) || !(y <= 2400000.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -9.5e+91) or not (y <= 2400000.0):
		tmp = y * ((z - x) / z)
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -9.5e+91) || !(y <= 2400000.0))
		tmp = Float64(y * Float64(Float64(z - x) / z));
	else
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -9.5e+91) || ~((y <= 2400000.0)))
		tmp = y * ((z - x) / z);
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -9.5e+91], N[Not[LessEqual[y, 2400000.0]], $MachinePrecision]], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5000000000000001e91 or 2.4e6 < y

   1. Initial program 78.2%

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

   3. Taylor expanded in y around inf 78.2%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   if -9.5000000000000001e91 < y < 2.4e6

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+91} \lor \neg \left(y \leq 2400000\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+142} \lor \neg \left(y \leq 2600000\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.5e+142) (not (<= y 2600000.0)))
   (* y (/ (- z x) z))
   (+ y (* x (/ (- 1.0 y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.5e+142) || !(y <= 2600000.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + (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) :: tmp
    if ((y <= (-5.5d+142)) .or. (.not. (y <= 2600000.0d0))) then
        tmp = y * ((z - x) / z)
    else
        tmp = y + (x * ((1.0d0 - y) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.5e+142) || !(y <= 2600000.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + (x * ((1.0 - y) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.5e+142) or not (y <= 2600000.0):
		tmp = y * ((z - x) / z)
	else:
		tmp = y + (x * ((1.0 - y) / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.5e+142) || !(y <= 2600000.0))
		tmp = Float64(y * Float64(Float64(z - x) / z));
	else
		tmp = Float64(y + Float64(x * Float64(Float64(1.0 - y) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.5e+142) || ~((y <= 2600000.0)))
		tmp = y * ((z - x) / z);
	else
		tmp = y + (x * ((1.0 - y) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.5e+142], N[Not[LessEqual[y, 2600000.0]], $MachinePrecision]], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.50000000000000035e142 or 2.6e6 < y

   1. Initial program 76.9%

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

   3. Taylor expanded in y around inf 76.9%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   if -5.50000000000000035e142 < y < 2.6e6

   1. Initial program 98.7%

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

      1. add-sqr-sqrt 54.7%

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

      2. pow2 54.7%

         \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt{z - x}\right)}^{2}}}{z} \]

   4. Applied egg-rr 54.7%

      \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt{z - x}\right)}^{2}}}{z} \]

   5. Taylor expanded in x around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. neg-mul-1 99.8%

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

      3. sub-neg 99.8%

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

      4. div-sub 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+142} \lor \neg \left(y \leq 2600000\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+23} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.3e+23) (not (<= y 1.0))) (* y (/ (- z x) z)) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.3e+23) || !(y <= 1.0)) {
		tmp = y * ((z - 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 <= (-2.3d+23)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * ((z - 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 <= -2.3e+23) || !(y <= 1.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.3e+23) or not (y <= 1.0):
		tmp = y * ((z - x) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.3e+23) || !(y <= 1.0))
		tmp = Float64(y * Float64(Float64(z - 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 <= -2.3e+23) || ~((y <= 1.0)))
		tmp = y * ((z - x) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.3e+23], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e23 or 1 < y

   1. Initial program 79.9%

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

   3. Taylor expanded in y around inf 79.0%

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

      1. associate-/l* 99.0%

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

   5. Simplified 99.0%

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

   if -2.3e23 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 97.4%

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

   4. Taylor expanded in x around 0 97.5%

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

      1. +-commutative 97.5%

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

   6. Simplified 97.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+23} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+25} \lor \neg \left(x \leq 2.15 \cdot 10^{+98}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.8e+25) (not (<= x 2.15e+98)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+25) || !(x <= 2.15e+98)) {
		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+25)) .or. (.not. (x <= 2.15d+98))) 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+25) || !(x <= 2.15e+98)) {
		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+25) or not (x <= 2.15e+98):
		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+25) || !(x <= 2.15e+98))
		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+25) || ~((x <= 2.15e+98)))
		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+25], N[Not[LessEqual[x, 2.15e+98]], $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 < -6.79999999999999967e25 or 2.1500000000000001e98 < x

   1. Initial program 93.5%

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

   3. Taylor expanded in x around inf 87.9%

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

      1. associate-/l* 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

   5. Simplified 90.1%

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

   if -6.79999999999999967e25 < x < 2.1500000000000001e98

   1. Initial program 87.7%

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

   3. Taylor expanded in z around inf 69.1%

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

   4. Taylor expanded in x around 0 81.3%

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

      1. +-commutative 81.3%

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

   6. Simplified 81.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+25} \lor \neg \left(x \leq 2.15 \cdot 10^{+98}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+150} \lor \neg \left(y \leq 1.55 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{y \cdot x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.45e+150) (not (<= y 1.55e+96)))
   (/ (* y x) (- z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.45e+150) || !(y <= 1.55e+96)) {
		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 <= (-1.45d+150)) .or. (.not. (y <= 1.55d+96))) 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 <= -1.45e+150) || !(y <= 1.55e+96)) {
		tmp = (y * x) / -z;
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.45e+150) or not (y <= 1.55e+96):
		tmp = (y * x) / -z
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.45e+150) || !(y <= 1.55e+96))
		tmp = Float64(Float64(y * x) / Float64(-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 <= -1.45e+150) || ~((y <= 1.55e+96)))
		tmp = (y * x) / -z;
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.45e+150], N[Not[LessEqual[y, 1.55e+96]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] / (-z)), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45000000000000005e150 or 1.5499999999999999e96 < y

   1. Initial program 74.8%

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

   3. Taylor expanded in x around inf 63.5%

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

      1. associate-/l* 53.5%

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

      2. mul-1-neg 53.5%

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

      3. unsub-neg 53.5%

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

   5. Simplified 53.5%

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

   6. Taylor expanded in y around inf 63.5%

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

      1. associate-*r/ 63.5%

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

      2. mul-1-neg 63.5%

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

      3. distribute-rgt-neg-out 63.5%

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

   8. Simplified 63.5%

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

   if -1.45000000000000005e150 < y < 1.5499999999999999e96

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf 88.4%

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

   4. Taylor expanded in x around 0 89.6%

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

      1. +-commutative 89.6%

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

   6. Simplified 89.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+150} \lor \neg \left(y \leq 1.55 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{y \cdot x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+217} \lor \neg \left(y \leq 1.8 \cdot 10^{+72}\right):\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.1e+217) (not (<= y 1.8e+72)))
   (* x (/ y (- z)))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.1e+217) || !(y <= 1.8e+72)) {
		tmp = 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 <= (-1.1d+217)) .or. (.not. (y <= 1.8d+72))) then
        tmp = 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 <= -1.1e+217) || !(y <= 1.8e+72)) {
		tmp = x * (y / -z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.1e+217) or not (y <= 1.8e+72):
		tmp = x * (y / -z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.1e+217) || !(y <= 1.8e+72))
		tmp = Float64(x * Float64(y / Float64(-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 <= -1.1e+217) || ~((y <= 1.8e+72)))
		tmp = x * (y / -z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.1e+217], N[Not[LessEqual[y, 1.8e+72]], $MachinePrecision]], N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e217 or 1.80000000000000017e72 < y

   1. Initial program 73.6%

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

   3. Taylor expanded in x around inf 62.1%

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

      1. associate-/l* 56.9%

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

      2. mul-1-neg 56.9%

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

      3. unsub-neg 56.9%

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

   5. Simplified 56.9%

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

   6. Taylor expanded in y around inf 62.1%

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

      1. mul-1-neg 62.1%

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

      2. associate-/l* 56.9%

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

      3. distribute-rgt-neg-in 56.9%

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

      4. distribute-neg-frac 56.9%

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

   8. Simplified 56.9%

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

   if -1.1e217 < y < 1.80000000000000017e72

   1. Initial program 96.7%

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

   3. Taylor expanded in z around inf 84.4%

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

   4. Taylor expanded in x around 0 87.1%

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

      1. +-commutative 87.1%

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

   6. Simplified 87.1%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+217} \lor \neg \left(y \leq 1.8 \cdot 10^{+72}\right):\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-34}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.85e-34) (* z (/ y z)) (if (<= y 7.2e-178) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.85e-34) {
		tmp = z * (y / z);
	} else if (y <= 7.2e-178) {
		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 <= (-1.85d-34)) then
        tmp = z * (y / z)
    else if (y <= 7.2d-178) 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 <= -1.85e-34) {
		tmp = z * (y / z);
	} else if (y <= 7.2e-178) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.85e-34:
		tmp = z * (y / z)
	elif y <= 7.2e-178:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.85e-34)
		tmp = Float64(z * Float64(y / z));
	elseif (y <= 7.2e-178)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.85e-34)
		tmp = z * (y / z);
	elseif (y <= 7.2e-178)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.85e-34], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-178], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 3 regimes

2. if y < -1.84999999999999994e-34

   1. Initial program 82.9%

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

   3. Taylor expanded in y around inf 82.9%

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

   4. Taylor expanded in z around inf 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-/l* 59.2%

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

   6. Applied egg-rr 59.2%

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

   if -1.84999999999999994e-34 < y < 7.19999999999999987e-178

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 79.1%

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

   if 7.19999999999999987e-178 < y

   1. Initial program 87.1%

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

   3. Taylor expanded in x around 0 47.2%

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

Alternative 9: 55.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-118}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 23500:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.3e-118) y (if (<= z 23500.0) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.3e-118) {
		tmp = y;
	} else if (z <= 23500.0) {
		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 <= (-2.3d-118)) then
        tmp = y
    else if (z <= 23500.0d0) 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 <= -2.3e-118) {
		tmp = y;
	} else if (z <= 23500.0) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.3e-118:
		tmp = y
	elif z <= 23500.0:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.3e-118)
		tmp = y;
	elseif (z <= 23500.0)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.3e-118)
		tmp = y;
	elseif (z <= 23500.0)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.3e-118], y, If[LessEqual[z, 23500.0], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if z < -2.30000000000000021e-118 or 23500 < z

   1. Initial program 82.3%

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

   3. Taylor expanded in x around 0 62.7%

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

   if -2.30000000000000021e-118 < z < 23500

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 53.8%

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

Alternative 10: 76.9% 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 90.1%

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

3. Taylor expanded in z around inf 65.2%

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

4. Taylor expanded in x around 0 73.2%

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

   1. +-commutative 73.2%

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

6. Simplified 73.2%

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

7. Final simplification 73.2%

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

Alternative 11: 40.2% 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 90.1%

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

3. Taylor expanded in x around 0 39.9%

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

Developer Target 1: 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 2024172 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ y (/ x z)) (/ y (/ z x))))

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