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

Percentage Accurate: 88.2% → 99.9%

Time: 10.8s

Alternatives: 8

Speedup: 1.1×

• 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: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

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

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. div-sub N/A

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

   14. unsub-neg N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-7}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ x z) y)))
   (if (<= y -3.9e+15) t_0 (if (<= y 1.08e-7) (+ y (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-y, (x / z), y);
	double tmp;
	if (y <= -3.9e+15) {
		tmp = t_0;
	} else if (y <= 1.08e-7) {
		tmp = y + (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-y), Float64(x / z), y)
	tmp = 0.0
	if (y <= -3.9e+15)
		tmp = t_0;
	elseif (y <= 1.08e-7)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-y) * N[(x / z), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -3.9e+15], t$95$0, If[LessEqual[y, 1.08e-7], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9e15 or 1.08000000000000001e-7 < y

   1. Initial program 75.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. sub-neg N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -3.9e15 < y < 1.08000000000000001e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.0%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity 99.0

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

      5. lift-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-lft-identity 99.0

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

   10. Applied rewrites 99.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-7}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-7}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (/ (* y x) z))))
   (if (<= y -3.9e+15) t_0 (if (<= y 1.08e-7) (+ y (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y - ((y * x) / z);
	double tmp;
	if (y <= -3.9e+15) {
		tmp = t_0;
	} else if (y <= 1.08e-7) {
		tmp = y + (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 - ((y * x) / z)
    if (y <= (-3.9d+15)) then
        tmp = t_0
    else if (y <= 1.08d-7) then
        tmp = y + (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 - ((y * x) / z);
	double tmp;
	if (y <= -3.9e+15) {
		tmp = t_0;
	} else if (y <= 1.08e-7) {
		tmp = y + (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y - ((y * x) / z)
	tmp = 0
	if y <= -3.9e+15:
		tmp = t_0
	elif y <= 1.08e-7:
		tmp = y + (x / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(Float64(y * x) / z))
	tmp = 0.0
	if (y <= -3.9e+15)
		tmp = t_0;
	elseif (y <= 1.08e-7)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y - ((y * x) / z);
	tmp = 0.0;
	if (y <= -3.9e+15)
		tmp = t_0;
	elseif (y <= 1.08e-7)
		tmp = y + (x / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.9e+15], t$95$0, If[LessEqual[y, 1.08e-7], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9e15 or 1.08000000000000001e-7 < y

   1. Initial program 75.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. sub-neg N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if -3.9e15 < y < 1.08000000000000001e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.0%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity 99.0

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

      5. lift-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-lft-identity 99.0

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

   10. Applied rewrites 99.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+15}:\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-7}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+130}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+199}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.5e+130)
   (+ y (/ x z))
   (if (<= y 2.5e+199) (* y (/ x (- z))) (* z (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.5e+130) {
		tmp = y + (x / z);
	} else if (y <= 2.5e+199) {
		tmp = y * (x / -z);
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5.5d+130) then
        tmp = y + (x / z)
    else if (y <= 2.5d+199) then
        tmp = y * (x / -z)
    else
        tmp = z * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.5e+130) {
		tmp = y + (x / z);
	} else if (y <= 2.5e+199) {
		tmp = y * (x / -z);
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.5e+130:
		tmp = y + (x / z)
	elif y <= 2.5e+199:
		tmp = y * (x / -z)
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.5e+130)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 2.5e+199)
		tmp = Float64(y * Float64(x / Float64(-z)));
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.5e+130)
		tmp = y + (x / z);
	elseif (y <= 2.5e+199)
		tmp = y * (x / -z);
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.5e+130], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+199], N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.4999999999999997e130

   1. Initial program 90.4%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.9%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity 84.9

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

      5. lift-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-lft-identity 84.9

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

   10. Applied rewrites 84.9%

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

   if 5.4999999999999997e130 < y < 2.4999999999999999e199

   1. Initial program 88.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 71.6

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

   8. Applied rewrites 71.6%

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

      1. lift-neg.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. remove-double-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 77.3

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

   10. Applied rewrites 77.3%

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

   if 2.4999999999999999e199 < y

   1. Initial program 44.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 20.3

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

   5. Applied rewrites 20.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 71.0

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

   7. Applied rewrites 71.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+130}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+199}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.1 \cdot 10^{+262}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.1e+262) (+ y (/ x z)) (* x (/ y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.1e+262) {
		tmp = y + (x / z);
	} else {
		tmp = x * (y / -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 <= 6.1d+262) then
        tmp = y + (x / z)
    else
        tmp = x * (y / -z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.1e+262) {
		tmp = y + (x / z);
	} else {
		tmp = x * (y / -z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 6.1e+262:
		tmp = y + (x / z)
	else:
		tmp = x * (y / -z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.1e+262)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(x * Float64(y / Float64(-z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 6.1e+262)
		tmp = y + (x / z);
	else
		tmp = x * (y / -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.1e+262], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.10000000000000031e262

   1. Initial program 86.7%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity 80.4

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

      5. lift-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-lft-identity 80.4

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

   10. Applied rewrites 80.4%

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

   if 6.10000000000000031e262 < x

   1. Initial program 88.1%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. sub-neg N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.1 \cdot 10^{+262}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-148}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.2e-148) y (if (<= y 2.1e-58) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2e-148) {
		tmp = y;
	} else if (y <= 2.1e-58) {
		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 (y <= (-1.2d-148)) then
        tmp = y
    else if (y <= 2.1d-58) 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 (y <= -1.2e-148) {
		tmp = y;
	} else if (y <= 2.1e-58) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.2e-148:
		tmp = y
	elif y <= 2.1e-58:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.2e-148)
		tmp = y;
	elseif (y <= 2.1e-58)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.2e-148)
		tmp = y;
	elseif (y <= 2.1e-58)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.2e-148], y, If[LessEqual[y, 2.1e-58], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2000000000000001e-148 or 2.09999999999999988e-58 < y

   1. Initial program 80.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 41.5

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

   5. Applied rewrites 41.5%

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

      1. associate-/l* N/A

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

      2. *-inverses N/A

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

      3. *-rgt-identity 57.0

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

   7. Applied rewrites 57.0%

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

   if -1.2000000000000001e-148 < y < 2.09999999999999988e-58

   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 77.3

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

   5. Applied rewrites 77.3%

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

Alternative 7: 77.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

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

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. div-sub N/A

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

   14. unsub-neg N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 79.2%

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

   1. lift-/.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lower-fma.f64 N/A

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

   4. *-lft-identity 79.2

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

   5. lift-fma.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. *-lft-identity 79.2

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

10. Applied rewrites 79.2%

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

11. Final simplification 79.2%

    \[\leadsto y + \frac{x}{z} \]
12. Add Preprocessing

Alternative 8: 41.1% 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 86.7%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 36.4

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

5. Applied rewrites 36.4%

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

   1. associate-/l* N/A

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

   2. *-inverses N/A

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

   3. *-rgt-identity 47.1

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

7. Applied rewrites 47.1%

   \[\leadsto \color{blue}{y} \]
8. Add Preprocessing

Developer Target 1: 93.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024216 
(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))