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

Percentage Accurate: 88.0% → 97.9%

Time: 9.7s

Alternatives: 10

Speedup: 0.9×

• 21.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.0% 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: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+35}:\\ \;\;\;\;y - \frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e+35) (- y (* (/ x z) y)) (fma (/ (- 1.0 y) z) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e+35) {
		tmp = y - ((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 <= -2e+35)
		tmp = Float64(y - Float64(Float64(x / z) * y));
	else
		tmp = fma(Float64(Float64(1.0 - y) / z), x, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e+35], N[(y - N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9999999999999999e35

   1. Initial program 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 73.0

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

   4. Applied rewrites 73.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower--.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -1.9999999999999999e35 < y

   1. Initial program 91.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 98.9%

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

Alternative 2: 51.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -11200:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-214}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+41}:\\ \;\;\;\;\frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -11200.0)
   (/ x z)
   (if (<= x -3e-214)
     (* (/ y x) x)
     (if (<= x 2.75e+41) (/ (* z y) z) (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -11200.0) {
		tmp = x / z;
	} else if (x <= -3e-214) {
		tmp = (y / x) * x;
	} else if (x <= 2.75e+41) {
		tmp = (z * y) / z;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-11200.0d0)) then
        tmp = x / z
    else if (x <= (-3d-214)) then
        tmp = (y / x) * x
    else if (x <= 2.75d+41) then
        tmp = (z * y) / z
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -11200.0) {
		tmp = x / z;
	} else if (x <= -3e-214) {
		tmp = (y / x) * x;
	} else if (x <= 2.75e+41) {
		tmp = (z * y) / z;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -11200.0:
		tmp = x / z
	elif x <= -3e-214:
		tmp = (y / x) * x
	elif x <= 2.75e+41:
		tmp = (z * y) / z
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -11200.0)
		tmp = Float64(x / z);
	elseif (x <= -3e-214)
		tmp = Float64(Float64(y / x) * x);
	elseif (x <= 2.75e+41)
		tmp = Float64(Float64(z * y) / z);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -11200.0)
		tmp = x / z;
	elseif (x <= -3e-214)
		tmp = (y / x) * x;
	elseif (x <= 2.75e+41)
		tmp = (z * y) / z;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -11200.0], N[(x / z), $MachinePrecision], If[LessEqual[x, -3e-214], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 2.75e+41], N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision], N[(x / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -11200 or 2.7500000000000002e41 < x

   1. Initial program 91.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 55.4

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

   5. Applied rewrites 55.4%

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

   if -11200 < x < -2.99999999999999994e-214

   1. Initial program 80.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 80.5

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

   4. Applied rewrites 80.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l- N/A

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

      9. div-sub N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 88.0

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

   7. Applied rewrites 88.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 60.0%

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

   if -2.99999999999999994e-214 < x < 2.7500000000000002e41

   1. Initial program 86.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.5

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

   5. Applied rewrites 52.5%

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

Alternative 3: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 78.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 25.0

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower--.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 98.3

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

   10. Applied rewrites 98.3%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.4%

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

   10. Applied rewrites 98.6%

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

   if 1 < y

   1. Initial program 75.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 33.7

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

   7. Applied rewrites 33.7%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower--.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.3

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

   10. Applied rewrites 99.3%

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

   12. Applied rewrites 99.4%

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

Alternative 4: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 76.3%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. 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 76.4

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 29.7

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower--.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 98.8

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

   10. Applied rewrites 98.8%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.4%

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

   10. Applied rewrites 98.6%

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

Alternative 5: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.2e+38)
   (* (/ (- 1.0 y) z) x)
   (if (<= x 5.2e+38) (fma 1.0 (/ x z) y) (* (- 1.0 y) (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.2e+38) {
		tmp = ((1.0 - y) / z) * x;
	} else if (x <= 5.2e+38) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = (1.0 - y) * (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.2e+38)
		tmp = Float64(Float64(Float64(1.0 - y) / z) * x);
	elseif (x <= 5.2e+38)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = Float64(Float64(1.0 - y) * Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.2e+38], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 5.2e+38], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2e38

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l* N/A

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

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

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. unsub-neg N/A

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

      18. mul-1-neg N/A

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

      19. lower-/.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 91.1

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

   5. Applied rewrites 91.1%

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

   if -4.2e38 < x < 5.1999999999999998e38

   1. Initial program 85.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.2%

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

   10. Applied rewrites 89.3%

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

   if 5.1999999999999998e38 < x

   1. Initial program 87.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 87.2

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. div-sub N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 91.1

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

   7. Applied rewrites 91.1%

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

Alternative 6: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - y}{z} \cdot x\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- 1.0 y) z) x)))
   (if (<= x -4.2e+38) t_0 (if (<= x 5.2e+38) (fma 1.0 (/ x z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 - y) / z) * x;
	double tmp;
	if (x <= -4.2e+38) {
		tmp = t_0;
	} else if (x <= 5.2e+38) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 - y) / z) * x)
	tmp = 0.0
	if (x <= -4.2e+38)
		tmp = t_0;
	elseif (x <= 5.2e+38)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.2e+38], t$95$0, If[LessEqual[x, 5.2e+38], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e38 or 5.1999999999999998e38 < x

   1. Initial program 90.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l* N/A

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

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

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. unsub-neg N/A

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

      18. mul-1-neg N/A

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

      19. lower-/.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -4.2e38 < x < 5.1999999999999998e38

   1. Initial program 85.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.2%

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

   10. Applied rewrites 89.3%

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

Alternative 7: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+165}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.2e+77)
   (fma 1.0 (/ x z) y)
   (if (<= y 2.55e+165) (* (/ (- y) z) x) (fma (/ 1.0 z) x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.2e+77) {
		tmp = fma(1.0, (x / z), y);
	} else if (y <= 2.55e+165) {
		tmp = (-y / z) * x;
	} else {
		tmp = fma((1.0 / z), x, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.2e+77)
		tmp = fma(1.0, Float64(x / z), y);
	elseif (y <= 2.55e+165)
		tmp = Float64(Float64(Float64(-y) / z) * x);
	else
		tmp = fma(Float64(1.0 / z), x, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.2e+77], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[y, 2.55e+165], N[(N[((-y) / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.1999999999999996e77

   1. Initial program 92.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.4%

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

   10. Applied rewrites 84.5%

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

   if 7.1999999999999996e77 < y < 2.5500000000000002e165

   1. Initial program 79.8%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 79.8

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. div-sub N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 84.3%

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

   if 2.5500000000000002e165 < y

   1. Initial program 62.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+165}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x} \cdot x\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ y x) x)))
   (if (<= z -2.4e+78) t_0 (if (<= z 8.6e+131) (/ x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y / x) * x;
	double tmp;
	if (z <= -2.4e+78) {
		tmp = t_0;
	} else if (z <= 8.6e+131) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (y / x) * x
    if (z <= (-2.4d+78)) then
        tmp = t_0
    else if (z <= 8.6d+131) then
        tmp = x / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / x) * x;
	double tmp;
	if (z <= -2.4e+78) {
		tmp = t_0;
	} else if (z <= 8.6e+131) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / x) * x
	tmp = 0
	if z <= -2.4e+78:
		tmp = t_0
	elif z <= 8.6e+131:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / x) * x)
	tmp = 0.0
	if (z <= -2.4e+78)
		tmp = t_0;
	elseif (z <= 8.6e+131)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / x) * x;
	tmp = 0.0;
	if (z <= -2.4e+78)
		tmp = t_0;
	elseif (z <= 8.6e+131)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -2.4e+78], t$95$0, If[LessEqual[z, 8.6e+131], N[(x / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3999999999999999e78 or 8.6000000000000003e131 < z

   1. Initial program 72.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 72.5

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l- N/A

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

      9. div-sub N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 91.0

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 59.8%

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

   if -2.3999999999999999e78 < z < 8.6000000000000003e131

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 48.9

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

   5. Applied rewrites 48.9%

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

Alternative 9: 78.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 76.1%

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

10. Applied rewrites 76.2%

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

Alternative 10: 39.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 38.7

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

5. Applied rewrites 38.7%

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

Developer Target 1: 94.0% 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 2024243 
(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))