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

Percentage Accurate: 88.1% → 99.9%

Time: 8.1s

Alternatives: 8

Speedup: 1.1×

• 20.2% 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: 88.1% 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 86.8%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) (- y) y)))
   (if (<= y -4.4e+18) t_0 (if (<= y 3.5e-6) (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 <= -4.4e+18) {
		tmp = t_0;
	} else if (y <= 3.5e-6) {
		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 <= -4.4e+18)
		tmp = t_0;
	elseif (y <= 3.5e-6)
		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, -4.4e+18], t$95$0, If[LessEqual[y, 3.5e-6], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4e18 or 3.49999999999999995e-6 < y

   1. Initial program 70.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 99.4%

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

   if -4.4e18 < y < 3.49999999999999995e-6

   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 98.9%

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

Alternative 3: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-y}{z}, x, y\right)\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ (- y) z) x y)))
   (if (<= y -4.4e+18) t_0 (if (<= y 3.5e-6) (fma (/ x z) 1.0 y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((-y / z), x, y);
	double tmp;
	if (y <= -4.4e+18) {
		tmp = t_0;
	} else if (y <= 3.5e-6) {
		tmp = fma((x / z), 1.0, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(Float64(-y) / z), x, y)
	tmp = 0.0
	if (y <= -4.4e+18)
		tmp = t_0;
	elseif (y <= 3.5e-6)
		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[((-y) / z), $MachinePrecision] * x + y), $MachinePrecision]}, If[LessEqual[y, -4.4e+18], t$95$0, If[LessEqual[y, 3.5e-6], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4e18 or 3.49999999999999995e-6 < y

   1. Initial program 70.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 90.0

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 89.5%

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

   if -4.4e18 < y < 3.49999999999999995e-6

   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 98.9%

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

Alternative 4: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{-96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-26}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) 1.0 y)))
   (if (<= z -6.2e-96) t_0 (if (<= z 1.4e-26) (* (- 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 <= -6.2e-96) {
		tmp = t_0;
	} else if (z <= 1.4e-26) {
		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 <= -6.2e-96)
		tmp = t_0;
	elseif (z <= 1.4e-26)
		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, -6.2e-96], t$95$0, If[LessEqual[z, 1.4e-26], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.1999999999999998e-96 or 1.4000000000000001e-26 < z

   1. Initial program 79.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 0

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

   8. Applied rewrites 86.7%

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

   if -6.1999999999999998e-96 < z < 1.4000000000000001e-26

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 90.1

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

   5. Applied rewrites 90.1%

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

   7. Applied rewrites 96.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-26}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{if}\;y \leq 9 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+281}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) 1.0 y)))
   (if (<= y 9e+19) t_0 (if (<= y 1.45e+281) (* (- 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 (y <= 9e+19) {
		tmp = t_0;
	} else if (y <= 1.45e+281) {
		tmp = -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 (y <= 9e+19)
		tmp = t_0;
	elseif (y <= 1.45e+281)
		tmp = Float64(Float64(-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[y, 9e+19], t$95$0, If[LessEqual[y, 1.45e+281], N[((-y) * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 9e19 or 1.45000000000000005e281 < y

   1. Initial program 91.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 87.8%

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

   if 9e19 < y < 1.45000000000000005e281

   1. Initial program 69.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 67.6%

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

   10. Applied rewrites 71.2%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+281}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-142}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.55e-142) (/ x z) (if (<= x 3.1e-36) (/ (* z y) z) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55e-142) {
		tmp = x / z;
	} else if (x <= 3.1e-36) {
		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 <= (-1.55d-142)) then
        tmp = x / z
    else if (x <= 3.1d-36) 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 <= -1.55e-142) {
		tmp = x / z;
	} else if (x <= 3.1e-36) {
		tmp = (z * y) / z;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.55e-142:
		tmp = x / z
	elif x <= 3.1e-36:
		tmp = (z * y) / z
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.55e-142)
		tmp = Float64(x / z);
	elseif (x <= 3.1e-36)
		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 <= -1.55e-142)
		tmp = x / z;
	elseif (x <= 3.1e-36)
		tmp = (z * y) / z;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.55e-142], N[(x / z), $MachinePrecision], If[LessEqual[x, 3.1e-36], N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision], N[(x / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55e-142 or 3.0999999999999999e-36 < x

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 49.6

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

   5. Applied rewrites 49.6%

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

   if -1.55e-142 < x < 3.0999999999999999e-36

   1. Initial program 84.7%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 58.2

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

   5. Applied rewrites 58.2%

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

Alternative 7: 78.3% 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 86.8%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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 75.9%

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

Alternative 8: 39.6% 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 86.8%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 40.1

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

5. Applied rewrites 40.1%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024294 
(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))