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

Percentage Accurate: 88.4% → 99.9%

Time: 7.0s

Alternatives: 6

Speedup: 1.0×

• 21.8% 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: 88.4% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ y + \left(1 - y\right) \cdot \frac{x}{z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return y + ((1.0 - 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 + ((1.0d0 - y) * (x / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

   1. +-commutative N/A

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

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

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

   3. associate-+l- N/A

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

   4. div-sub N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. fmm-def N/A

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

   8. *-inverses N/A

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

   9. fma-define N/A

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

   10. *-rgt-identity N/A

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

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y - x}{z}\right)\right)}\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\mathsf{neg}\left(\left(x \cdot y - x\right)\right)}{\color{blue}{z}}\right)\right) \]

   13. sub0-neg N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{0 - \left(x \cdot y - x\right)}{z}\right)\right) \]

   14. associate-+l- N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(0 - x \cdot y\right) + x}{z}\right)\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x \cdot y\right)\right) + x}{z}\right)\right) \]

   16. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y + x}{z}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right) + x}{z}\right)\right) \]

   18. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(-1 \cdot x\right) + x}{z}\right)\right) \]

   19. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1\right) \cdot x + x}{z}\right)\right) \]

   20. distribute-lft1-in N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1 + 1\right) \cdot x}{z}\right)\right) \]

   21. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\left(y \cdot -1 + 1\right) \cdot \color{blue}{\frac{x}{z}}\right)\right) \]

   22. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot -1 + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

3. Simplified 100.0%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (/ x z)))))
   (if (<= y -2.6e+28) t_0 (if (<= y 1.0) (+ y (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - (x / z));
	double tmp;
	if (y <= -2.6e+28) {
		tmp = t_0;
	} else if (y <= 1.0) {
		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 * (1.0d0 - (x / z))
    if (y <= (-2.6d+28)) then
        tmp = t_0
    else if (y <= 1.0d0) 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 * (1.0 - (x / z));
	double tmp;
	if (y <= -2.6e+28) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - (x / z))
	tmp = 0
	if y <= -2.6e+28:
		tmp = t_0
	elif y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - Float64(x / z)))
	tmp = 0.0
	if (y <= -2.6e+28)
		tmp = t_0;
	elseif (y <= 1.0)
		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 * (1.0 - (x / z));
	tmp = 0.0;
	if (y <= -2.6e+28)
		tmp = t_0;
	elseif (y <= 1.0)
		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[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.6e+28], t$95$0, If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6000000000000002e28 or 1 < y

   1. Initial program 75.3%

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

      1. +-commutative N/A

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

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

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

      3. associate-+l- N/A

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

      4. div-sub N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. fmm-def N/A

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

      8. *-inverses N/A

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

      9. fma-define N/A

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

      10. *-rgt-identity N/A

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

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y - x}{z}\right)\right)}\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\mathsf{neg}\left(\left(x \cdot y - x\right)\right)}{\color{blue}{z}}\right)\right) \]

      13. sub0-neg N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{0 - \left(x \cdot y - x\right)}{z}\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(0 - x \cdot y\right) + x}{z}\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x \cdot y\right)\right) + x}{z}\right)\right) \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y + x}{z}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right) + x}{z}\right)\right) \]

      18. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(-1 \cdot x\right) + x}{z}\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1\right) \cdot x + x}{z}\right)\right) \]

      20. distribute-lft1-in N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1 + 1\right) \cdot x}{z}\right)\right) \]

      21. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\left(y \cdot -1 + 1\right) \cdot \color{blue}{\frac{x}{z}}\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot -1 + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-inverses N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{z - x}{z}\right)}\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{z}{z} - \color{blue}{\frac{x}{z}}\right)\right) \]

      7. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(1 - \frac{\color{blue}{x}}{z}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

      9. /-lowering-/.f64 97.7%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   7. Simplified 97.7%

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

   if -2.6000000000000002e28 < y < 1

   1. Initial program 98.5%

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

      1. +-commutative N/A

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

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

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

      3. associate-+l- N/A

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

      4. div-sub N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. fmm-def N/A

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

      8. *-inverses N/A

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

      9. fma-define N/A

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

      10. *-rgt-identity N/A

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

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y - x}{z}\right)\right)}\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\mathsf{neg}\left(\left(x \cdot y - x\right)\right)}{\color{blue}{z}}\right)\right) \]

      13. sub0-neg N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{0 - \left(x \cdot y - x\right)}{z}\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(0 - x \cdot y\right) + x}{z}\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x \cdot y\right)\right) + x}{z}\right)\right) \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y + x}{z}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right) + x}{z}\right)\right) \]

      18. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(-1 \cdot x\right) + x}{z}\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1\right) \cdot x + x}{z}\right)\right) \]

      20. distribute-lft1-in N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1 + 1\right) \cdot x}{z}\right)\right) \]

      21. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\left(y \cdot -1 + 1\right) \cdot \color{blue}{\frac{x}{z}}\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot -1 + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\frac{x}{z}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 98.7%

         \[\leadsto \mathsf{+.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Simplified 98.7%

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

Alternative 3: 62.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -3 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ y z))))
   (if (<= y -3e-16) t_0 (if (<= y 7.5e-54) (/ x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y / z);
	double tmp;
	if (y <= -3e-16) {
		tmp = t_0;
	} else if (y <= 7.5e-54) {
		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 = z * (y / z)
    if (y <= (-3d-16)) then
        tmp = t_0
    else if (y <= 7.5d-54) 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 = z * (y / z);
	double tmp;
	if (y <= -3e-16) {
		tmp = t_0;
	} else if (y <= 7.5e-54) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y / z)
	tmp = 0
	if y <= -3e-16:
		tmp = t_0
	elif y <= 7.5e-54:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y / z))
	tmp = 0.0
	if (y <= -3e-16)
		tmp = t_0;
	elseif (y <= 7.5e-54)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y / z);
	tmp = 0.0;
	if (y <= -3e-16)
		tmp = t_0;
	elseif (y <= 7.5e-54)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e-16], t$95$0, If[LessEqual[y, 7.5e-54], N[(x / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.99999999999999994e-16 or 7.5000000000000005e-54 < y

   1. Initial program 77.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot z\right)}, z\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot y\right), z\right) \]

      2. *-lowering-*.f64 34.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, y\right), z\right) \]

   5. Simplified 34.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{z}\right), \color{blue}{z}\right) \]

      4. /-lowering-/.f64 52.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), z\right) \]

   7. Applied egg-rr 52.7%

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

   if -2.99999999999999994e-16 < y < 7.5000000000000005e-54

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

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

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

      3. associate-+l- N/A

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

      4. div-sub N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. fmm-def N/A

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

      8. *-inverses N/A

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

      9. fma-define N/A

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

      10. *-rgt-identity N/A

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

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y - x}{z}\right)\right)}\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\mathsf{neg}\left(\left(x \cdot y - x\right)\right)}{\color{blue}{z}}\right)\right) \]

      13. sub0-neg N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{0 - \left(x \cdot y - x\right)}{z}\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(0 - x \cdot y\right) + x}{z}\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x \cdot y\right)\right) + x}{z}\right)\right) \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y + x}{z}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right) + x}{z}\right)\right) \]

      18. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(-1 \cdot x\right) + x}{z}\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1\right) \cdot x + x}{z}\right)\right) \]

      20. distribute-lft1-in N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1 + 1\right) \cdot x}{z}\right)\right) \]

      21. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\left(y \cdot -1 + 1\right) \cdot \color{blue}{\frac{x}{z}}\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot -1 + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 74.4%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   7. Simplified 74.4%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e-16) y (if (<= y 3.9e-54) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e-16) {
		tmp = y;
	} else if (y <= 3.9e-54) {
		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 <= (-3.1d-16)) then
        tmp = y
    else if (y <= 3.9d-54) 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 <= -3.1e-16) {
		tmp = y;
	} else if (y <= 3.9e-54) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.1e-16:
		tmp = y
	elif y <= 3.9e-54:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e-16)
		tmp = y;
	elseif (y <= 3.9e-54)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.1e-16)
		tmp = y;
	elseif (y <= 3.9e-54)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.1e-16], y, If[LessEqual[y, 3.9e-54], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1000000000000001e-16 or 3.9e-54 < y

   1. Initial program 77.5%

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

      1. +-commutative N/A

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

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

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

      3. associate-+l- N/A

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

      4. div-sub N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. fmm-def N/A

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

      8. *-inverses N/A

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

      9. fma-define N/A

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

      10. *-rgt-identity N/A

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

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y - x}{z}\right)\right)}\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\mathsf{neg}\left(\left(x \cdot y - x\right)\right)}{\color{blue}{z}}\right)\right) \]

      13. sub0-neg N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{0 - \left(x \cdot y - x\right)}{z}\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(0 - x \cdot y\right) + x}{z}\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x \cdot y\right)\right) + x}{z}\right)\right) \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y + x}{z}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right) + x}{z}\right)\right) \]

      18. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(-1 \cdot x\right) + x}{z}\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1\right) \cdot x + x}{z}\right)\right) \]

      20. distribute-lft1-in N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1 + 1\right) \cdot x}{z}\right)\right) \]

      21. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\left(y \cdot -1 + 1\right) \cdot \color{blue}{\frac{x}{z}}\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot -1 + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 51.1%

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

   if -3.1000000000000001e-16 < y < 3.9e-54

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

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

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

      3. associate-+l- N/A

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

      4. div-sub N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. fmm-def N/A

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

      8. *-inverses N/A

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

      9. fma-define N/A

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

      10. *-rgt-identity N/A

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

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y - x}{z}\right)\right)}\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\mathsf{neg}\left(\left(x \cdot y - x\right)\right)}{\color{blue}{z}}\right)\right) \]

      13. sub0-neg N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{0 - \left(x \cdot y - x\right)}{z}\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(0 - x \cdot y\right) + x}{z}\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x \cdot y\right)\right) + x}{z}\right)\right) \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y + x}{z}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right) + x}{z}\right)\right) \]

      18. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(-1 \cdot x\right) + x}{z}\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1\right) \cdot x + x}{z}\right)\right) \]

      20. distribute-lft1-in N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1 + 1\right) \cdot x}{z}\right)\right) \]

      21. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(y, \left(\left(y \cdot -1 + 1\right) \cdot \color{blue}{\frac{x}{z}}\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot -1 + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 74.4%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   7. Simplified 74.4%

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

Alternative 5: 78.1% accurate, 1.8× 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 87.5%

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

   1. +-commutative N/A

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

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

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

   3. associate-+l- N/A

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

   4. div-sub N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. fmm-def N/A

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

   8. *-inverses N/A

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

   9. fma-define N/A

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

   10. *-rgt-identity N/A

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

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y - x}{z}\right)\right)}\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\mathsf{neg}\left(\left(x \cdot y - x\right)\right)}{\color{blue}{z}}\right)\right) \]

   13. sub0-neg N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{0 - \left(x \cdot y - x\right)}{z}\right)\right) \]

   14. associate-+l- N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(0 - x \cdot y\right) + x}{z}\right)\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x \cdot y\right)\right) + x}{z}\right)\right) \]

   16. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y + x}{z}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right) + x}{z}\right)\right) \]

   18. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(-1 \cdot x\right) + x}{z}\right)\right) \]

   19. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1\right) \cdot x + x}{z}\right)\right) \]

   20. distribute-lft1-in N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1 + 1\right) \cdot x}{z}\right)\right) \]

   21. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\left(y \cdot -1 + 1\right) \cdot \color{blue}{\frac{x}{z}}\right)\right) \]

   22. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot -1 + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\frac{x}{z}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 77.5%

      \[\leadsto \mathsf{+.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

7. Simplified 77.5%

   \[\leadsto y + \color{blue}{\frac{x}{z}} \]
8. Add Preprocessing

Alternative 6: 40.5% accurate, 9.0× 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 87.5%

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

   1. +-commutative N/A

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

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

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

   3. associate-+l- N/A

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

   4. div-sub N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. fmm-def N/A

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

   8. *-inverses N/A

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

   9. fma-define N/A

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

   10. *-rgt-identity N/A

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

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y - x}{z}\right)\right)}\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\mathsf{neg}\left(\left(x \cdot y - x\right)\right)}{\color{blue}{z}}\right)\right) \]

   13. sub0-neg N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{0 - \left(x \cdot y - x\right)}{z}\right)\right) \]

   14. associate-+l- N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(0 - x \cdot y\right) + x}{z}\right)\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x \cdot y\right)\right) + x}{z}\right)\right) \]

   16. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y + x}{z}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right) + x}{z}\right)\right) \]

   18. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{y \cdot \left(-1 \cdot x\right) + x}{z}\right)\right) \]

   19. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1\right) \cdot x + x}{z}\right)\right) \]

   20. distribute-lft1-in N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\frac{\left(y \cdot -1 + 1\right) \cdot x}{z}\right)\right) \]

   21. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(y, \left(\left(y \cdot -1 + 1\right) \cdot \color{blue}{\frac{x}{z}}\right)\right) \]

   22. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot -1 + 1\right), \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 40.4%

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

Developer Target 1: 94.0% accurate, 0.8× 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 2024158 
(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))