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

Percentage Accurate: 87.8% → 99.9%

Time: 10.0s

Alternatives: 8

Speedup: 1.1×

• 21.4% 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: 87.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. div-sub N/A

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

   14. unsub-neg N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{-z}, y, y\right)\\ \mathbf{if}\;y \leq -1020000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0142:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x (- z)) y y)))
   (if (<= y -1020000000.0) t_0 (if (<= y 0.0142) (fma 1.0 (/ x z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / -z), y, y);
	double tmp;
	if (y <= -1020000000.0) {
		tmp = t_0;
	} else if (y <= 0.0142) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / Float64(-z)), y, y)
	tmp = 0.0
	if (y <= -1020000000.0)
		tmp = t_0;
	elseif (y <= 0.0142)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / (-z)), $MachinePrecision] * y + y), $MachinePrecision]}, If[LessEqual[y, -1020000000.0], t$95$0, If[LessEqual[y, 0.0142], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02e9 or 0.014200000000000001 < y

   1. Initial program 76.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. sub-neg N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   7. Applied rewrites 99.8%

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

   if -1.02e9 < y < 0.014200000000000001

   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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.7%

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

Alternative 3: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \frac{x \cdot y}{z}\\ \mathbf{if}\;y \leq -1020000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0142:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (/ (* x y) z))))
   (if (<= y -1020000000.0) t_0 (if (<= y 0.0142) (fma 1.0 (/ x z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y - ((x * y) / z);
	double tmp;
	if (y <= -1020000000.0) {
		tmp = t_0;
	} else if (y <= 0.0142) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(Float64(x * y) / z))
	tmp = 0.0
	if (y <= -1020000000.0)
		tmp = t_0;
	elseif (y <= 0.0142)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1020000000.0], t$95$0, If[LessEqual[y, 0.0142], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02e9 or 0.014200000000000001 < y

   1. Initial program 76.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. sub-neg N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if -1.02e9 < y < 0.014200000000000001

   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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1020000000:\\ \;\;\;\;y - \frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 0.0142:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1 - y}{z}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ (- 1.0 y) z))))
   (if (<= x -1.15e+22) t_0 (if (<= x 4.5e+41) (fma 1.0 (/ x z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * ((1.0 - y) / z);
	double tmp;
	if (x <= -1.15e+22) {
		tmp = t_0;
	} else if (x <= 4.5e+41) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(1.0 - y) / z))
	tmp = 0.0
	if (x <= -1.15e+22)
		tmp = t_0;
	elseif (x <= 4.5e+41)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e+22], t$95$0, If[LessEqual[x, 4.5e+41], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1500000000000001e22 or 4.5000000000000001e41 < x

   1. Initial program 91.2%

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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in x around inf

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

       1. associate-/l* N/A

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

       2. mul-1-neg N/A

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

       3. sub-neg N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. mul-1-neg N/A

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

       7. lower-/.f64 N/A

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

       8. mul-1-neg N/A

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

       9. sub-neg N/A

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

       10. lower--.f64 88.6

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

   11. Applied rewrites 88.6%

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

   if -1.1500000000000001e22 < x < 4.5000000000000001e41

   1. Initial program 86.6%

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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 91.0%

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

Alternative 5: 51.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{y \cdot z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.08e+25) (/ x z) (if (<= x 1.1e-55) (/ (* y z) z) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.08e+25) {
		tmp = x / z;
	} else if (x <= 1.1e-55) {
		tmp = (y * z) / 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 <= (-1.08d+25)) then
        tmp = x / z
    else if (x <= 1.1d-55) then
        tmp = (y * z) / 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 <= -1.08e+25) {
		tmp = x / z;
	} else if (x <= 1.1e-55) {
		tmp = (y * z) / z;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.08e+25:
		tmp = x / z
	elif x <= 1.1e-55:
		tmp = (y * z) / z
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.08e+25)
		tmp = Float64(x / z);
	elseif (x <= 1.1e-55)
		tmp = Float64(Float64(y * z) / z);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.08e+25)
		tmp = x / z;
	elseif (x <= 1.1e-55)
		tmp = (y * z) / z;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.08e+25], N[(x / z), $MachinePrecision], If[LessEqual[x, 1.1e-55], N[(N[(y * z), $MachinePrecision] / z), $MachinePrecision], N[(x / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.08e25 or 1.1e-55 < x

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 50.4

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

   5. Applied rewrites 50.4%

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

   if -1.08e25 < x < 1.1e-55

   1. Initial program 85.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

Alternative 6: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e+225) (* x (/ (- y) z)) (fma 1.0 (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e+225) {
		tmp = x * (-y / z);
	} else {
		tmp = fma(1.0, (x / z), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e+225)
		tmp = Float64(x * Float64(Float64(-y) / z));
	else
		tmp = fma(1.0, Float64(x / z), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4e+225], N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e225

   1. Initial program 88.4%

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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 40.5%

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

   9. Taylor expanded in x around inf

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

       1. associate-/l* N/A

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

       2. mul-1-neg N/A

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

       3. sub-neg N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. mul-1-neg N/A

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

       7. lower-/.f64 N/A

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

       8. mul-1-neg N/A

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

       9. sub-neg N/A

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

       10. lower--.f64 97.9

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

   11. Applied rewrites 97.9%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 73.3%

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

   if -1.4e225 < x

   1. Initial program 88.6%

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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.7%

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

Alternative 7: 78.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. div-sub N/A

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

   14. unsub-neg N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 79.2%

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

Alternative 8: 39.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 35.8

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

5. Applied rewrites 35.8%

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

Developer Target 1: 93.5% 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 2024238 
(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))