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

Percentage Accurate: 88.1% → 99.9%

Time: 5.2s

Alternatives: 9

Speedup: 1.1×

• 20.9% 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: 88.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + (y * (z - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \mathbf{if}\;y \leq -680:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) (- y) y)))
   (if (<= y -680.0) t_0 (if (<= y 1.0) (fma (/ x z) 1.0 y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), -y, y);
	double tmp;
	if (y <= -680.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma((x / z), 1.0, 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 <= -680.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(Float64(x / z), 1.0, 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, -680.0], t$95$0, If[LessEqual[y, 1.0], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -680 or 1 < y

   1. Initial program 74.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.2%

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

   if -680 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.6%

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

Alternative 3: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(z - x\right) \cdot y}{z}\\ \mathbf{if}\;y \leq -680:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (- z x) y) z)))
   (if (<= y -680.0) t_0 (if (<= y 1.0) (fma (/ x z) 1.0 y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((z - x) * y) / z;
	double tmp;
	if (y <= -680.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma((x / z), 1.0, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z - x) * y) / z)
	tmp = 0.0
	if (y <= -680.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(Float64(x / z), 1.0, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -680.0], t$95$0, If[LessEqual[y, 1.0], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -680 or 1 < y

   1. Initial program 74.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if -680 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.6%

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

Alternative 4: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{if}\;z \leq -2.25 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) 1.0 y)))
   (if (<= z -2.25e-72) t_0 (if (<= z 3.8e-106) (* (- 1.0 y) (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), 1.0, y);
	double tmp;
	if (z <= -2.25e-72) {
		tmp = t_0;
	} else if (z <= 3.8e-106) {
		tmp = (1.0 - y) * (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), 1.0, y)
	tmp = 0.0
	if (z <= -2.25e-72)
		tmp = t_0;
	elseif (z <= 3.8e-106)
		tmp = Float64(Float64(1.0 - y) * Float64(x / z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision]}, If[LessEqual[z, -2.25e-72], t$95$0, If[LessEqual[z, 3.8e-106], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.25e-72 or 3.7999999999999999e-106 < z

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.2%

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

   if -2.25e-72 < z < 3.7999999999999999e-106

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 76.6

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 87.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot y}{z}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z y) z)))
   (if (<= y -1.15e-51) t_0 (if (<= y 6.2e-124) (/ x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * y) / z;
	double tmp;
	if (y <= -1.15e-51) {
		tmp = t_0;
	} else if (y <= 6.2e-124) {
		tmp = 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 * y) / z
    if (y <= (-1.15d-51)) then
        tmp = t_0
    else if (y <= 6.2d-124) then
        tmp = 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 * y) / z;
	double tmp;
	if (y <= -1.15e-51) {
		tmp = t_0;
	} else if (y <= 6.2e-124) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * y) / z
	tmp = 0
	if y <= -1.15e-51:
		tmp = t_0
	elif y <= 6.2e-124:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * y) / z)
	tmp = 0.0
	if (y <= -1.15e-51)
		tmp = t_0;
	elseif (y <= 6.2e-124)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * y) / z;
	tmp = 0.0;
	if (y <= -1.15e-51)
		tmp = t_0;
	elseif (y <= 6.2e-124)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -1.15e-51], t$95$0, If[LessEqual[y, 6.2e-124], N[(x / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15000000000000001e-51 or 6.1999999999999996e-124 < y

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 41.0

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

   5. Applied rewrites 41.0%

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

   if -1.15000000000000001e-51 < y < 6.1999999999999996e-124

   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 81.9

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

   5. Applied rewrites 81.9%

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

Alternative 6: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.3e+187) (fma (/ x z) 1.0 y) (* (/ (- y) z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.3e+187) {
		tmp = fma((x / z), 1.0, y);
	} else {
		tmp = (-y / z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.3e+187)
		tmp = fma(Float64(x / z), 1.0, y);
	else
		tmp = Float64(Float64(Float64(-y) / z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.3e+187], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], N[(N[((-y) / z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3000000000000001e187

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.9%

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

   if 3.3000000000000001e187 < x

   1. Initial program 68.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 78.2%

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

Alternative 7: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.5e+187) (fma (/ x z) 1.0 y) (* (- y) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.5e+187) {
		tmp = fma((x / z), 1.0, y);
	} else {
		tmp = -y * (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.5e+187)
		tmp = fma(Float64(x / z), 1.0, y);
	else
		tmp = Float64(Float64(-y) * Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.5e+187], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], N[((-y) * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5000000000000001e187

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.9%

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

   if 2.5000000000000001e187 < x

   1. Initial program 68.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 78.2%

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

   10. Applied rewrites 78.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 76.9%

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

Alternative 9: 40.4% 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 87.1%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 35.2

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

5. Applied rewrites 35.2%

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

Developer Target 1: 94.3% 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 2024308 
(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))