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

Percentage Accurate: 88.5% → 99.0%

Time: 9.1s

Alternatives: 9

Speedup: 1.1×

• 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: 88.5% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z - x}{z} \cdot y\\ \mathbf{if}\;y \leq -0.88:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.96:\\ \;\;\;\;\frac{z \cdot y + x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- z x) z) y)))
   (if (<= y -0.88) t_0 (if (<= y 0.96) (/ (+ (* z y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((z - x) / z) * y;
	double tmp;
	if (y <= -0.88) {
		tmp = t_0;
	} else if (y <= 0.96) {
		tmp = ((z * y) + x) / z;
	} else {
		tmp = t_0;
	}
	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 = ((z - x) / z) * y
    if (y <= (-0.88d0)) then
        tmp = t_0
    else if (y <= 0.96d0) then
        tmp = ((z * y) + x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((z - x) / z) * y;
	double tmp;
	if (y <= -0.88) {
		tmp = t_0;
	} else if (y <= 0.96) {
		tmp = ((z * y) + x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((z - x) / z) * y
	tmp = 0
	if y <= -0.88:
		tmp = t_0
	elif y <= 0.96:
		tmp = ((z * y) + x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z - x) / z) * y)
	tmp = 0.0
	if (y <= -0.88)
		tmp = t_0;
	elseif (y <= 0.96)
		tmp = Float64(Float64(Float64(z * y) + x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((z - x) / z) * y;
	tmp = 0.0;
	if (y <= -0.88)
		tmp = t_0;
	elseif (y <= 0.96)
		tmp = ((z * y) + x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -0.88], t$95$0, If[LessEqual[y, 0.96], N[(N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.880000000000000004 or 0.95999999999999996 < y

   1. Initial program 70.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 97.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if -0.880000000000000004 < y < 0.95999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.88:\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{elif}\;y \leq 0.96:\\ \;\;\;\;\frac{z \cdot y + x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z - x}{z} \cdot y\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.96:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- z x) z) y)))
   (if (<= y -3.3e+19) t_0 (if (<= y 0.96) (fma (/ 1.0 z) x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((z - x) / z) * y;
	double tmp;
	if (y <= -3.3e+19) {
		tmp = t_0;
	} else if (y <= 0.96) {
		tmp = fma((1.0 / z), x, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z - x) / z) * y)
	tmp = 0.0
	if (y <= -3.3e+19)
		tmp = t_0;
	elseif (y <= 0.96)
		tmp = fma(Float64(1.0 / z), x, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -3.3e+19], t$95$0, If[LessEqual[y, 0.96], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e19 or 0.95999999999999996 < y

   1. Initial program 70.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 96.9%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if -3.3e19 < y < 0.95999999999999996

   1. Initial program 99.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 99.8%

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

   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 99.8%

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

Alternative 3: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{if}\;z \leq -9 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ 1.0 z) x y)))
   (if (<= z -9e-21) t_0 (if (<= z 8.3e-38) (* (/ (- 1.0 y) z) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((1.0 / z), x, y);
	double tmp;
	if (z <= -9e-21) {
		tmp = t_0;
	} else if (z <= 8.3e-38) {
		tmp = ((1.0 - y) / z) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(1.0 / z), x, y)
	tmp = 0.0
	if (z <= -9e-21)
		tmp = t_0;
	elseif (z <= 8.3e-38)
		tmp = Float64(Float64(Float64(1.0 - y) / z) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision]}, If[LessEqual[z, -9e-21], t$95$0, If[LessEqual[z, 8.3e-38], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999936e-21 or 8.2999999999999995e-38 < z

   1. Initial program 74.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 99.9%

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

   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 86.1%

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

   if -8.99999999999999936e-21 < z < 8.2999999999999995e-38

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x + -1 \cdot \left(x \cdot y\right)}{z}} \]
   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 89.2

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

   5. Applied rewrites 89.2%

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

Alternative 4: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-x\right) \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e+228) (fma (/ 1.0 z) x y) (/ (* (- x) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e+228) {
		tmp = fma((1.0 / z), x, y);
	} else {
		tmp = (-x * y) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.2e+228)
		tmp = fma(Float64(1.0 / z), x, y);
	else
		tmp = Float64(Float64(Float64(-x) * y) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.2e+228], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision], N[(N[((-x) * y), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.2e228

   1. Initial program 85.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 98.7%

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

   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 80.3%

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

   if 8.2e228 < y

   1. Initial program 83.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.1%

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

Alternative 5: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e+228) (fma (/ 1.0 z) x y) (* (/ (- y) z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e+228) {
		tmp = fma((1.0 / z), x, y);
	} else {
		tmp = (-y / z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e+228)
		tmp = fma(Float64(1.0 / z), x, y);
	else
		tmp = Float64(Float64(Float64(-y) / z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e+228], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision], N[(N[((-y) / z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000002e228

   1. Initial program 85.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 98.7%

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

   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 80.3%

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

   if 8.5000000000000002e228 < y

   1. Initial program 83.1%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x + -1 \cdot \left(x \cdot y\right)}{z}} \]
   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 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.1%

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

Alternative 6: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-44}:\\ \;\;\;\;1 \cdot y\\ \mathbf{elif}\;y \leq 0.013:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.5e-44) (* 1.0 y) (if (<= y 0.013) (/ x z) (* 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e-44) {
		tmp = 1.0 * y;
	} else if (y <= 0.013) {
		tmp = x / z;
	} else {
		tmp = 1.0 * 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 <= (-1.5d-44)) then
        tmp = 1.0d0 * y
    else if (y <= 0.013d0) then
        tmp = x / z
    else
        tmp = 1.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e-44) {
		tmp = 1.0 * y;
	} else if (y <= 0.013) {
		tmp = x / z;
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.5e-44:
		tmp = 1.0 * y
	elif y <= 0.013:
		tmp = x / z
	else:
		tmp = 1.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.5e-44)
		tmp = Float64(1.0 * y);
	elseif (y <= 0.013)
		tmp = Float64(x / z);
	else
		tmp = Float64(1.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.5e-44)
		tmp = 1.0 * y;
	elseif (y <= 0.013)
		tmp = x / z;
	else
		tmp = 1.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.5e-44], N[(1.0 * y), $MachinePrecision], If[LessEqual[y, 0.013], N[(x / z), $MachinePrecision], N[(1.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5000000000000001e-44 or 0.0129999999999999994 < y

   1. Initial program 72.6%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 97.2%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 97.8

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

   8. Applied rewrites 97.8%

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

   9. Taylor expanded in z around inf

      \[\leadsto 1 \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 53.0%

       \[\leadsto 1 \cdot y \]

   if -1.5000000000000001e-44 < y < 0.0129999999999999994

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

Alternative 7: 95.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.9%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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 98.3%

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

Alternative 8: 77.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.9%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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 98.3%

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

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 77.4%

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

Alternative 9: 40.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 1.0 y))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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 98.3%

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

6. Taylor expanded in y around inf

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 68.0

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

8. Applied rewrites 68.0%

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

9. Taylor expanded in z around inf

   \[\leadsto 1 \cdot y \]
10. Step-by-step derivation

11. Applied rewrites 41.0%

    \[\leadsto 1 \cdot y \]
12. Add Preprocessing

Developer Target 1: 94.1% 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 2024249 
(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))