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

Percentage Accurate: 88.8% → 99.9%

Time: 7.9s

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: 88.8% 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 87.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 87.4

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

4. Applied rewrites 87.4%

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

5. Taylor expanded in x around 0

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

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. associate-/l* N/A

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

   11. lower-fma.f64 N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -320 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -320.0) (not (<= y 1.0)))
   (fma (- y) (/ x z) y)
   (fma 1.0 (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -320.0) || !(y <= 1.0)) {
		tmp = fma(-y, (x / z), y);
	} else {
		tmp = fma(1.0, (x / z), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -320.0) || !(y <= 1.0))
		tmp = fma(Float64(-y), Float64(x / z), y);
	else
		tmp = fma(1.0, Float64(x / z), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -320.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[((-y) * N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -320 or 1 < y

   1. Initial program 72.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 72.6

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in x around 0

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

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 99.5%

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

   if -320 < y < 1

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

   10. 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 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -320 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-10} \lor \neg \left(x \leq 5.3 \cdot 10^{+126}\right):\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.4e-10) (not (<= x 5.3e+126)))
   (* (- 1.0 y) (/ x z))
   (fma 1.0 (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.4e-10) || !(x <= 5.3e+126)) {
		tmp = (1.0 - y) * (x / z);
	} else {
		tmp = fma(1.0, (x / z), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.4e-10) || !(x <= 5.3e+126))
		tmp = Float64(Float64(1.0 - y) * Float64(x / z));
	else
		tmp = fma(1.0, Float64(x / z), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.4e-10], N[Not[LessEqual[x, 5.3e+126]], $MachinePrecision]], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3999999999999998e-10 or 5.30000000000000028e126 < x

   1. Initial program 89.6%

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. neg-mul-1 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -4.3999999999999998e-10 < x < 5.30000000000000028e126

   1. Initial program 86.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 86.0

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

   4. Applied rewrites 86.0%

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

   5. Taylor expanded in x around 0

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

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 92.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-10} \lor \neg \left(x \leq 5.3 \cdot 10^{+126}\right):\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -320:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, x, y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -320.0)
   (fma (/ (- y) z) x y)
   (if (<= y 1.0) (fma 1.0 (/ x z) y) (fma (- y) (/ x z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -320.0) {
		tmp = fma((-y / z), x, y);
	} else if (y <= 1.0) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = fma(-y, (x / z), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -320.0)
		tmp = fma(Float64(Float64(-y) / z), x, y);
	elseif (y <= 1.0)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = fma(Float64(-y), Float64(x / z), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -320.0], N[(N[((-y) / z), $MachinePrecision] * x + y), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[((-y) * N[(x / z), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -320

   1. Initial program 73.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 73.1

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in x around 0

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

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 99.1%

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

   12. Applied rewrites 99.2%

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

   if -320 < y < 1

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.7%

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

   if 1 < y

   1. Initial program 72.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 72.1

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in x around 0

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

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 99.9%

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

Alternative 5: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-10}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.4e-10)
   (* (- 1.0 y) (/ x z))
   (if (<= x 6.7e+137) (fma 1.0 (/ x z) y) (* (/ (- 1.0 y) z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.4e-10) {
		tmp = (1.0 - y) * (x / z);
	} else if (x <= 6.7e+137) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = ((1.0 - y) / z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.4e-10)
		tmp = Float64(Float64(1.0 - y) * Float64(x / z));
	elseif (x <= 6.7e+137)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = Float64(Float64(Float64(1.0 - y) / z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.4e-10], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.7e+137], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3999999999999998e-10

   1. Initial program 86.6%

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. neg-mul-1 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -4.3999999999999998e-10 < x < 6.6999999999999999e137

   1. Initial program 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 86.1

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

   4. Applied rewrites 86.1%

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

   5. Taylor expanded in x around 0

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

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 92.4%

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

   if 6.6999999999999999e137 < x

   1. Initial program 96.7%

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. neg-mul-1 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-10}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.92 \cdot 10^{-22} \lor \neg \left(y \leq 0.0044\right):\\ \;\;\;\;\frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.92e-22) (not (<= y 0.0044))) (/ (* z y) z) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.92e-22) || !(y <= 0.0044)) {
		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 ((y <= (-1.92d-22)) .or. (.not. (y <= 0.0044d0))) 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 ((y <= -1.92e-22) || !(y <= 0.0044)) {
		tmp = (z * y) / z;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.92e-22) or not (y <= 0.0044):
		tmp = (z * y) / z
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.92e-22) || !(y <= 0.0044))
		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 ((y <= -1.92e-22) || ~((y <= 0.0044)))
		tmp = (z * y) / z;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.92e-22], N[Not[LessEqual[y, 0.0044]], $MachinePrecision]], N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.91999999999999997e-22 or 0.00440000000000000027 < y

   1. Initial program 74.7%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 38.7

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

   5. Applied rewrites 38.7%

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

   if -1.91999999999999997e-22 < y < 0.00440000000000000027

   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 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.92 \cdot 10^{-22} \lor \neg \left(y \leq 0.0044\right):\\ \;\;\;\;\frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 9.2e+244) (fma 1.0 (/ x z) y) (* (- y) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.2e+244) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = -y * (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 9.2e+244)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = Float64(Float64(-y) * Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 9.2e+244], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[((-y) * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.1999999999999998e244

   1. Initial program 87.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 87.2

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in x around 0

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

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 83.9%

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

   if 9.1999999999999998e244 < x

   1. Initial program 92.4%

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. neg-mul-1 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 91.7%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.1% 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 87.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 87.4

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

4. Applied rewrites 87.4%

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

5. Taylor expanded in x around 0

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

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. associate-/l* N/A

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

   11. lower-fma.f64 N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 81.3%

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

Alternative 9: 40.2% 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 87.4%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 40.8

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

5. Applied rewrites 40.8%

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

6. Final simplification 40.8%

   \[\leadsto \frac{x}{z} \]
7. Add Preprocessing

Developer Target 1: 93.9% 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 2024296 
(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))