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

Percentage Accurate: 87.9% → 99.2%

Time: 7.6s

Alternatives: 9

Speedup: 0.7×

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: 87.9% 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: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+81} \lor \neg \left(z \leq 1.85 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + z \cdot y\right) - x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.4e+81) (not (<= z 1.85e-75)))
   (fma (/ (- 1.0 y) z) x y)
   (/ (- (+ x (* z y)) (* x y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.4e+81) || !(z <= 1.85e-75)) {
		tmp = fma(((1.0 - y) / z), x, y);
	} else {
		tmp = ((x + (z * y)) - (x * y)) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.4e+81) || !(z <= 1.85e-75))
		tmp = fma(Float64(Float64(1.0 - y) / z), x, y);
	else
		tmp = Float64(Float64(Float64(x + Float64(z * y)) - Float64(x * y)) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.4e+81], N[Not[LessEqual[z, 1.85e-75]], $MachinePrecision]], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x + y), $MachinePrecision], N[(N[(N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.40000000000000003e81 or 1.85000000000000012e-75 < z

   1. Initial program 80.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 80.2

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

   4. Applied rewrites 80.2%

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

   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. distribute-rgt-in N/A

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

      3. distribute-rgt-in N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-/l* N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

   7. Applied rewrites 99.9%

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

   if -3.40000000000000003e81 < z < 1.85000000000000012e-75

   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. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+81} \lor \neg \left(z \leq 1.85 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + z \cdot y\right) - x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+81} \lor \neg \left(z \leq 1.85 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.4e+81) (not (<= z 1.85e-75)))
   (fma (/ (- 1.0 y) z) x y)
   (/ (fma (- z x) y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.4e+81) || !(z <= 1.85e-75)) {
		tmp = fma(((1.0 - y) / z), x, y);
	} else {
		tmp = fma((z - x), y, x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.4e+81) || !(z <= 1.85e-75))
		tmp = fma(Float64(Float64(1.0 - y) / z), x, y);
	else
		tmp = Float64(fma(Float64(z - x), y, x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.4e+81], N[Not[LessEqual[z, 1.85e-75]], $MachinePrecision]], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x + y), $MachinePrecision], N[(N[(N[(z - x), $MachinePrecision] * y + x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.40000000000000003e81 or 1.85000000000000012e-75 < z

   1. Initial program 80.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 80.2

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

   4. Applied rewrites 80.2%

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

   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. distribute-rgt-in N/A

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

      3. distribute-rgt-in N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-/l* N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

   7. Applied rewrites 99.9%

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

   if -3.40000000000000003e81 < z < 1.85000000000000012e-75

   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%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+81} \lor \neg \left(z \leq 1.85 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+104} \lor \neg \left(y \leq 10^{+19}\right):\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.45e+104) (not (<= y 1e+19)))
   (* (/ (- z x) z) y)
   (fma (/ (- 1.0 y) z) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.45e+104) || !(y <= 1e+19)) {
		tmp = ((z - x) / z) * y;
	} else {
		tmp = fma(((1.0 - y) / z), x, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.45e+104) || !(y <= 1e+19))
		tmp = Float64(Float64(Float64(z - x) / z) * y);
	else
		tmp = fma(Float64(Float64(1.0 - y) / z), x, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.45e+104], N[Not[LessEqual[y, 1e+19]], $MachinePrecision]], N[(N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4499999999999999e104 or 1e19 < y

   1. Initial program 69.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 69.2

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

   4. Applied rewrites 69.2%

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

   5. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -1.4499999999999999e104 < y < 1e19

   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. distribute-rgt-in N/A

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

      3. distribute-rgt-in N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-/l* N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+104} \lor \neg \left(y \leq 10^{+19}\right):\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8500000 \lor \neg \left(y \leq 0.032\right):\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8500000.0) (not (<= y 0.032)))
   (* (/ (- z x) z) y)
   (fma (/ 1.0 z) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8500000.0) || !(y <= 0.032)) {
		tmp = ((z - x) / z) * y;
	} else {
		tmp = fma((1.0 / z), x, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8500000.0) || !(y <= 0.032))
		tmp = Float64(Float64(Float64(z - x) / z) * y);
	else
		tmp = fma(Float64(1.0 / z), x, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8500000.0], N[Not[LessEqual[y, 0.032]], $MachinePrecision]], N[(N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5e6 or 0.032000000000000001 < y

   1. Initial program 75.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 75.4

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if -8.5e6 < y < 0.032000000000000001

   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. distribute-rgt-in N/A

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

      3. distribute-rgt-in N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-/l* N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8500000 \lor \neg \left(y \leq 0.032\right):\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+90} \lor \neg \left(x \leq 2.2 \cdot 10^{+139}\right):\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.7e+90) (not (<= x 2.2e+139)))
   (* (- 1.0 y) (/ x z))
   (fma (/ 1.0 z) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.7e+90) || !(x <= 2.2e+139)) {
		tmp = (1.0 - y) * (x / z);
	} else {
		tmp = fma((1.0 / z), x, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.7e+90) || !(x <= 2.2e+139))
		tmp = Float64(Float64(1.0 - y) * Float64(x / z));
	else
		tmp = fma(Float64(1.0 / z), x, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.7e+90], N[Not[LessEqual[x, 2.2e+139]], $MachinePrecision]], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000001e90 or 2.1999999999999999e139 < x

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      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 94.1

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

   5. Applied rewrites 94.1%

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

   if -4.7000000000000001e90 < x < 2.1999999999999999e139

   1. Initial program 85.8%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 85.8

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

   4. Applied rewrites 85.8%

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

   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. distribute-rgt-in N/A

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

      3. distribute-rgt-in N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-/l* N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 90.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+90} \lor \neg \left(x \leq 2.2 \cdot 10^{+139}\right):\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+90}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.7e+90)
   (* (- 1.0 y) (/ x z))
   (if (<= x 5.5e+139) (fma (/ 1.0 z) x y) (* (/ (- 1.0 y) z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.7e+90) {
		tmp = (1.0 - y) * (x / z);
	} else if (x <= 5.5e+139) {
		tmp = fma((1.0 / z), x, y);
	} else {
		tmp = ((1.0 - y) / z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.7e+90)
		tmp = Float64(Float64(1.0 - y) * Float64(x / z));
	elseif (x <= 5.5e+139)
		tmp = fma(Float64(1.0 / z), x, y);
	else
		tmp = Float64(Float64(Float64(1.0 - y) / z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.7e+90], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e+139], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7000000000000001e90

   1. Initial program 97.3%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   if -4.7000000000000001e90 < x < 5.4999999999999996e139

   1. Initial program 85.8%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 85.8

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

   4. Applied rewrites 85.8%

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

   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. distribute-rgt-in N/A

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

      3. distribute-rgt-in N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. mul-1-neg N/A

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

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

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

      9. associate-/l* N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 90.8%

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

   if 5.4999999999999996e139 < x

   1. Initial program 94.6%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in x around inf

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

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 88.2

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

   7. Applied rewrites 88.2%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+90}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+21} \lor \neg \left(y \leq 1.6 \cdot 10^{-22}\right):\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.1e+21) (not (<= y 1.6e-22))) (* 1.0 y) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.1e+21) || !(y <= 1.6e-22)) {
		tmp = 1.0 * y;
	} 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 <= (-1.1d+21)) .or. (.not. (y <= 1.6d-22))) then
        tmp = 1.0d0 * y
    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 <= -1.1e+21) || !(y <= 1.6e-22)) {
		tmp = 1.0 * y;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.1e+21) or not (y <= 1.6e-22):
		tmp = 1.0 * y
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.1e+21) || !(y <= 1.6e-22))
		tmp = Float64(1.0 * y);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.1e+21) || ~((y <= 1.6e-22)))
		tmp = 1.0 * y;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.1e+21], N[Not[LessEqual[y, 1.6e-22]], $MachinePrecision]], N[(1.0 * y), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e21 or 1.59999999999999994e-22 < y

   1. Initial program 76.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 76.5

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 96.6

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

   7. Applied rewrites 96.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.6%

       \[\leadsto 1 \cdot y \]

   if -1.1e21 < y < 1.59999999999999994e-22

   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 71.9

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

   5. Applied rewrites 71.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+21} \lor \neg \left(y \leq 1.6 \cdot 10^{-22}\right):\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.6%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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 88.6

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

4. Applied rewrites 88.6%

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

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. distribute-rgt-in N/A

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

   3. distribute-rgt-in N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. mul-1-neg N/A

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

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

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

   9. associate-/l* N/A

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

   10. mul-1-neg N/A

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

   11. +-commutative N/A

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

7. Applied rewrites 95.4%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 84.7%

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

11. Final simplification 84.7%

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

Alternative 9: 41.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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 88.6

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

4. Applied rewrites 88.6%

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

5. Taylor expanded in y around inf

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 65.4

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

7. Applied rewrites 65.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 45.7%

    \[\leadsto 1 \cdot y \]

11. Final simplification 45.7%

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

Developer Target 1: 94.6% 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 2024313 
(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))