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

Percentage Accurate: 88.7% → 99.9%

Time: 6.7s

Alternatives: 8

Speedup: 1.1×

• 21.5% 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: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. lower--.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.42\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.42)))
   (fma (/ x z) (- y) y)
   (fma (/ x z) 1.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.42)) {
		tmp = fma((x / z), -y, y);
	} else {
		tmp = fma((x / z), 1.0, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.42))
		tmp = fma(Float64(x / z), Float64(-y), y);
	else
		tmp = fma(Float64(x / z), 1.0, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.42]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * (-y) + y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.419999999999999984 < y

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.9%

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

   if -1 < y < 0.419999999999999984

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.42\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.42\right):\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.42)))
   (* (/ (- z x) z) y)
   (fma (/ x z) 1.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.42)) {
		tmp = ((z - x) / z) * y;
	} else {
		tmp = fma((x / z), 1.0, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.42))
		tmp = Float64(Float64(Float64(z - x) / z) * y);
	else
		tmp = fma(Float64(x / z), 1.0, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.42]], $MachinePrecision]], N[(N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.419999999999999984 < y

   1. Initial program 76.9%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-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 99.8

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

   5. Applied rewrites 99.8%

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

   if -1 < y < 0.419999999999999984

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.42\right):\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+191} \lor \neg \left(x \leq 1.3 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.8e+191) (not (<= x 1.3e+91)))
   (* (/ (- 1.0 y) z) x)
   (fma (/ x z) 1.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.8e+191) || !(x <= 1.3e+91)) {
		tmp = ((1.0 - y) / z) * x;
	} else {
		tmp = fma((x / z), 1.0, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.8e+191) || !(x <= 1.3e+91))
		tmp = Float64(Float64(Float64(1.0 - y) / z) * x);
	else
		tmp = fma(Float64(x / z), 1.0, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.8e+191], N[Not[LessEqual[x, 1.3e+91]], $MachinePrecision]], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999972e191 or 1.3e91 < x

   1. Initial program 88.6%

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

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if -4.79999999999999972e191 < x < 1.3e91

   1. Initial program 89.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 86.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+191} \lor \neg \left(x \leq 1.3 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+262}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+17)
   (fma (/ x z) 1.0 y)
   (if (<= y 3.2e+262) (* (/ x z) (- y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+17) {
		tmp = fma((x / z), 1.0, y);
	} else if (y <= 3.2e+262) {
		tmp = (x / z) * -y;
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+17)
		tmp = fma(Float64(x / z), 1.0, y);
	elseif (y <= 3.2e+262)
		tmp = Float64(Float64(x / z) * Float64(-y));
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+17], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], If[LessEqual[y, 3.2e+262], N[(N[(x / z), $MachinePrecision] * (-y)), $MachinePrecision], y]]

Derivation

1. Split input into 3 regimes

2. if y < 2e17

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.6%

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

   if 2e17 < y < 3.1999999999999998e262

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.8%

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

   10. Applied rewrites 63.2%

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

   if 3.1999999999999998e262 < y

   1. Initial program 62.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. unpow1 N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{{\left(y \cdot \left(z - x\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + x}{z} \]

      5. sqrt-pow1 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{{\left(y \cdot \left(z - x\right)\right)}^{2}}} + x}{z} \]

      6. pow2 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 25.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 25.7

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

   4. Applied rewrites 25.7%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot y\right)} \]

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot y \]

      6. *-lft-identity 67.1

         \[\leadsto \color{blue}{y} \]

   7. Applied rewrites 67.1%

      \[\leadsto \color{blue}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+262}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -155:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -155.0) y (if (<= z 2.9e-86) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -155.0) {
		tmp = y;
	} else if (z <= 2.9e-86) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	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 (z <= (-155.0d0)) then
        tmp = y
    else if (z <= 2.9d-86) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -155.0) {
		tmp = y;
	} else if (z <= 2.9e-86) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -155.0:
		tmp = y
	elif z <= 2.9e-86:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -155.0)
		tmp = y;
	elseif (z <= 2.9e-86)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -155.0)
		tmp = y;
	elseif (z <= 2.9e-86)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -155.0], y, If[LessEqual[z, 2.9e-86], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if z < -155 or 2.8999999999999999e-86 < z

   1. Initial program 81.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. unpow1 N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{{\left(y \cdot \left(z - x\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + x}{z} \]

      5. sqrt-pow1 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{{\left(y \cdot \left(z - x\right)\right)}^{2}}} + x}{z} \]

      6. pow2 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 42.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot y\right)} \]

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot y \]

      6. *-lft-identity 67.8

         \[\leadsto \color{blue}{y} \]

   7. Applied rewrites 67.8%

      \[\leadsto \color{blue}{y} \]

   if -155 < z < 2.8999999999999999e-86

   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 58.3

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

   5. Applied rewrites 58.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -155:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. lower--.f64 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 79.7%

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

Alternative 8: 40.3% accurate, 23.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 89.3%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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. unpow1 N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{{\left(y \cdot \left(z - x\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + x}{z} \]

   5. sqrt-pow1 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{{\left(y \cdot \left(z - x\right)\right)}^{2}}} + x}{z} \]

   6. pow2 N/A

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

   7. sqrt-prod N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sqrt.f64 46.6

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 46.6

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

4. Applied rewrites 46.6%

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

5. Taylor expanded in z around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto -1 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot y\right)} \]

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

      \[\leadsto \color{blue}{1} \cdot y \]

   6. *-lft-identity 44.7

      \[\leadsto \color{blue}{y} \]

7. Applied rewrites 44.7%

   \[\leadsto \color{blue}{y} \]

8. Final simplification 44.7%

   \[\leadsto y \]
9. Add Preprocessing

Developer Target 1: 93.6% 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 2024320 
(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))