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

Percentage Accurate: 88.2% → 99.2%

Time: 9.0s

Alternatives: 8

Speedup: 0.9×

• 20.5% 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: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \mathbf{if}\;y \leq -210:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) (- y) y)))
   (if (<= y -210.0) t_0 (if (<= y 1.0) (fma 1.0 (/ x z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), -y, y);
	double tmp;
	if (y <= -210.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), Float64(-y), y)
	tmp = 0.0
	if (y <= -210.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * (-y) + y), $MachinePrecision]}, If[LessEqual[y, -210.0], t$95$0, If[LessEqual[y, 1.0], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -210 or 1 < y

   1. Initial program 74.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 91.7%

      \[\leadsto \mathsf{fma}\left(\frac{-y}{z}, x, y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

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

   if -210 < 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 0

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.8%

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.1%

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

Alternative 2: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - y}{z} \cdot x\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- 1.0 y) z) x)))
   (if (<= x -6.8e+18) t_0 (if (<= x 1.46e-12) (fma 1.0 (/ x z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 - y) / z) * x;
	double tmp;
	if (x <= -6.8e+18) {
		tmp = t_0;
	} else if (x <= 1.46e-12) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 - y) / z) * x)
	tmp = 0.0
	if (x <= -6.8e+18)
		tmp = t_0;
	elseif (x <= 1.46e-12)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6.8e+18], t$95$0, If[LessEqual[x, 1.46e-12], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8e18 or 1.46000000000000004e-12 < x

   1. Initial program 88.2%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l* N/A

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

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

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. unsub-neg N/A

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

      18. mul-1-neg N/A

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

      19. lower-/.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if -6.8e18 < x < 1.46000000000000004e-12

   1. Initial program 88.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.5%

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 91.7%

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

Alternative 3: 51.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-142}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45e-142) (/ x z) (if (<= x 2.9e-108) (/ (* z y) z) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-142) {
		tmp = x / z;
	} else if (x <= 2.9e-108) {
		tmp = (z * y) / z;
	} else {
		tmp = 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 <= (-1.45d-142)) then
        tmp = x / z
    else if (x <= 2.9d-108) then
        tmp = (z * y) / z
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-142) {
		tmp = x / z;
	} else if (x <= 2.9e-108) {
		tmp = (z * y) / z;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.45e-142:
		tmp = x / z
	elif x <= 2.9e-108:
		tmp = (z * y) / z
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45e-142)
		tmp = Float64(x / z);
	elseif (x <= 2.9e-108)
		tmp = Float64(Float64(z * y) / z);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.45e-142)
		tmp = x / z;
	elseif (x <= 2.9e-108)
		tmp = (z * y) / z;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45e-142], N[(x / z), $MachinePrecision], If[LessEqual[x, 2.9e-108], N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision], N[(x / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999995e-142 or 2.9000000000000001e-108 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if -1.44999999999999995e-142 < x < 2.9000000000000001e-108

   1. Initial program 88.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

Alternative 4: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+24) (fma (/ (- 1.0 y) z) x y) (fma (/ x z) (- y) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+24) {
		tmp = fma(((1.0 - y) / z), x, y);
	} else {
		tmp = fma((x / z), -y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+24)
		tmp = fma(Float64(Float64(1.0 - y) / z), x, y);
	else
		tmp = fma(Float64(x / z), Float64(-y), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e+24], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x + y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * (-y) + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999998e23

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 98.4%

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

   if 9.9999999999999998e23 < y

   1. Initial program 69.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 89.0%

      \[\leadsto \mathsf{fma}\left(\frac{-y}{z}, x, y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

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

Alternative 5: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5e+67) (fma 1.0 (/ x z) y) (* (- y) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e+67) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = -y * (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5e+67)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = Float64(Float64(-y) * Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5e+67], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[((-y) * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.49999999999999988e67

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 90.0%

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 90.1%

       \[\leadsto \mathsf{fma}\left(1, \color{blue}{\frac{x}{z}}, y\right) \]

   if 2.49999999999999988e67 < y

   1. Initial program 72.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 60.7%

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

Alternative 6: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5e+67) (fma 1.0 (/ x z) y) (* (/ (- y) z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e+67) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = (-y / z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5e+67)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = Float64(Float64(Float64(-y) / z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5e+67], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[(N[((-y) / z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.49999999999999988e67

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 90.0%

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 90.1%

       \[\leadsto \mathsf{fma}\left(1, \color{blue}{\frac{x}{z}}, y\right) \]

   if 2.49999999999999988e67 < y

   1. Initial program 72.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 54.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, \frac{x}{z}, y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma 1.0 (/ x z) y))

C

double code(double x, double y, double z) {
	return fma(1.0, (x / z), y);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 96.5%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
7. Step-by-step derivation

8. Applied rewrites 80.6%

   \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
9. Step-by-step derivation

10. Applied rewrites 80.7%

    \[\leadsto \mathsf{fma}\left(1, \color{blue}{\frac{x}{z}}, y\right) \]
11. Add Preprocessing

Alternative 8: 39.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 43.9

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

5. Applied rewrites 43.9%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
6. Add Preprocessing

Developer Target 1: 94.0% accurate, 0.6× 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 2024235 
(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))