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

Percentage Accurate: 88.5% → 100.0%

Time: 11.0s

Alternatives: 7

Speedup: 1.1×

• 20.9% 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: 88.5% 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: 100.0% 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 84.9%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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. Simplified 99.9%

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

Alternative 2: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * (-y) + y), $MachinePrecision]}, If[LessEqual[y, -1.0], 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 < -1 or 1 < y

   1. Initial program 70.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. Simplified 99.9%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 98.5

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

   8. Simplified 98.5%

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

   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 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. Simplified 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. Simplified 99.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{x}{z}, y\right) \]
   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 99.3

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

      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 99.3

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

   10. Applied egg-rr 99.3%

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

4. Final simplification 98.9%

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

Alternative 3: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (/ (* y x) z))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (+ 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 <= -1.0) {
		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 - ((y * x) / z)
    if (y <= (-1.0d0)) 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 - ((y * x) / z);
	double tmp;
	if (y <= -1.0) {
		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 - ((y * x) / z)
	tmp = 0
	if y <= -1.0:
		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(Float64(y * x) / z))
	tmp = 0.0
	if (y <= -1.0)
		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 - ((y * x) / z);
	tmp = 0.0;
	if (y <= -1.0)
		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[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], 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 < -1 or 1 < y

   1. Initial program 70.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 91.7

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

   5. Simplified 91.7%

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

   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 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. Simplified 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. Simplified 99.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{x}{z}, y\right) \]
   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 99.3

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

      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 99.3

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

   10. Applied egg-rr 99.3%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;z \leq -0.00065:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-136}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))))
   (if (<= z -0.00065) t_0 (if (<= z 4.3e-136) (* (- 1.0 y) (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (z <= -0.00065) {
		tmp = t_0;
	} else if (z <= 4.3e-136) {
		tmp = (1.0 - 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 + (x / z)
    if (z <= (-0.00065d0)) then
        tmp = t_0
    else if (z <= 4.3d-136) then
        tmp = (1.0d0 - 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 + (x / z);
	double tmp;
	if (z <= -0.00065) {
		tmp = t_0;
	} else if (z <= 4.3e-136) {
		tmp = (1.0 - y) * (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x / z)
	tmp = 0
	if z <= -0.00065:
		tmp = t_0
	elif z <= 4.3e-136:
		tmp = (1.0 - y) * (x / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x / z))
	tmp = 0.0
	if (z <= -0.00065)
		tmp = t_0;
	elseif (z <= 4.3e-136)
		tmp = Float64(Float64(1.0 - y) * Float64(x / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (x / z);
	tmp = 0.0;
	if (z <= -0.00065)
		tmp = t_0;
	elseif (z <= 4.3e-136)
		tmp = (1.0 - y) * (x / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.00065], t$95$0, If[LessEqual[z, 4.3e-136], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4999999999999997e-4 or 4.3e-136 < z

   1. Initial program 75.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. Simplified 99.9%

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

   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. Simplified 89.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{x}{z}, y\right) \]
   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 89.1

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

      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 89.1

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

   10. Applied egg-rr 89.1%

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

   if -6.4999999999999997e-4 < z < 4.3e-136

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 87.6

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

   5. Simplified 87.6%

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

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

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 87.6

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

   7. Applied egg-rr 87.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00065:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-136}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-74}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e-74) y (if (<= z 3.3e+18) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e-74) {
		tmp = y;
	} else if (z <= 3.3e+18) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-4.8d-74)) then
        tmp = y
    else if (z <= 3.3d+18) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e-74) {
		tmp = y;
	} else if (z <= 3.3e+18) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.8e-74:
		tmp = y
	elif z <= 3.3e+18:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e-74)
		tmp = y;
	elseif (z <= 3.3e+18)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.8e-74)
		tmp = y;
	elseif (z <= 3.3e+18)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e-74], y, If[LessEqual[z, 3.3e+18], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if z < -4.7999999999999998e-74 or 3.3e18 < z

   1. Initial program 74.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{3} + {\left(y \cdot \left(z - x\right)\right)}^{3}}{x \cdot x + \left(\left(y \cdot \left(z - x\right)\right) \cdot \left(y \cdot \left(z - x\right)\right) - x \cdot \left(y \cdot \left(z - x\right)\right)\right)}}}{z} \]

      4. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot \left(z - x\right)\right) \cdot \left(y \cdot \left(z - x\right)\right) - x \cdot \left(y \cdot \left(z - x\right)\right)\right)}{{x}^{3} + {\left(y \cdot \left(z - x\right)\right)}^{3}}}}}{z} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot \left(z - x\right)\right) \cdot \left(y \cdot \left(z - x\right)\right) - x \cdot \left(y \cdot \left(z - x\right)\right)\right)}{{x}^{3} + {\left(y \cdot \left(z - x\right)\right)}^{3}}}}}{z} \]

      6. clear-num N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(y \cdot \left(z - x\right)\right)}^{3}}{x \cdot x + \left(\left(y \cdot \left(z - x\right)\right) \cdot \left(y \cdot \left(z - x\right)\right) - x \cdot \left(y \cdot \left(z - x\right)\right)\right)}}}}}{z} \]

      7. flip3-+ N/A

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

      8. lift-+.f64 N/A

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

      9. lower-/.f64 74.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 74.3

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

   4. Applied egg-rr 74.3%

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

   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 47.1

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

   7. Simplified 47.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity 67.7

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

   9. Applied egg-rr 67.7%

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

   if -4.7999999999999998e-74 < z < 3.3e18

   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 62.3

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

   5. Simplified 62.3%

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

Alternative 6: 78.1% 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 84.9%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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. Simplified 99.9%

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

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. Simplified 80.8%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{x}{z}, y\right) \]
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 80.8

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

   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 80.8

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

10. Applied egg-rr 80.8%

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

11. Final simplification 80.8%

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

Alternative 7: 40.4% 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 84.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. flip3-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{{x}^{3} + {\left(y \cdot \left(z - x\right)\right)}^{3}}{x \cdot x + \left(\left(y \cdot \left(z - x\right)\right) \cdot \left(y \cdot \left(z - x\right)\right) - x \cdot \left(y \cdot \left(z - x\right)\right)\right)}}}{z} \]

   4. clear-num N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot \left(z - x\right)\right) \cdot \left(y \cdot \left(z - x\right)\right) - x \cdot \left(y \cdot \left(z - x\right)\right)\right)}{{x}^{3} + {\left(y \cdot \left(z - x\right)\right)}^{3}}}}}{z} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot \left(z - x\right)\right) \cdot \left(y \cdot \left(z - x\right)\right) - x \cdot \left(y \cdot \left(z - x\right)\right)\right)}{{x}^{3} + {\left(y \cdot \left(z - x\right)\right)}^{3}}}}}{z} \]

   6. clear-num N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(y \cdot \left(z - x\right)\right)}^{3}}{x \cdot x + \left(\left(y \cdot \left(z - x\right)\right) \cdot \left(y \cdot \left(z - x\right)\right) - x \cdot \left(y \cdot \left(z - x\right)\right)\right)}}}}}{z} \]

   7. flip3-+ N/A

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

   8. lift-+.f64 N/A

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

   9. lower-/.f64 84.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 84.8

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

4. Applied egg-rr 84.8%

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

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 33.6

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

7. Simplified 33.6%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-inverses N/A

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

   4. *-rgt-identity 45.7

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

9. Applied egg-rr 45.7%

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

Developer Target 1: 93.7% 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 2024207 
(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))