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

Percentage Accurate: 88.5% → 99.9%

Time: 10.4s

Alternatives: 7

Speedup: 1.1×

• 21.3% 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: 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 82.7%

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

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

   11. +-commutative N/A

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

   12. mul-1-neg N/A

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

   13. unsub-neg N/A

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

   14. div-sub N/A

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

   15. unsub-neg N/A

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

   16. mul-1-neg N/A

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

5. Simplified 100.0%

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

Alternative 2: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{if}\;y \leq -280000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ x z) y)))
   (if (<= y -280000.0) t_0 (if (<= y 1.0) (+ y (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-y, (x / z), y);
	double tmp;
	if (y <= -280000.0) {
		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(-y), Float64(x / z), y)
	tmp = 0.0
	if (y <= -280000.0)
		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[((-y) * N[(x / z), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -280000.0], 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 < -2.8e5 or 1 < y

   1. Initial program 70.0%

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

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. div-sub N/A

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

      15. unsub-neg N/A

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

      16. mul-1-neg N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.3

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

   8. Simplified 99.3%

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

   if -2.8e5 < 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

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. div-sub N/A

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

      15. unsub-neg N/A

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

      16. mul-1-neg N/A

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

   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 98.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 98.4

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

   11. Simplified 98.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -280000:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -280000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (/ (* y x) z))))
   (if (<= y -280000.0) 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 <= -280000.0) {
		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 <= (-280000.0d0)) 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 <= -280000.0) {
		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 <= -280000.0:
		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 <= -280000.0)
		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 <= -280000.0)
		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, -280000.0], 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 < -2.8e5 or 1 < y

   1. Initial program 70.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 87.8

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

   5. Simplified 87.8%

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

   if -2.8e5 < 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

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. div-sub N/A

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

      15. unsub-neg N/A

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

      16. mul-1-neg N/A

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

   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 98.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 98.4

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

   11. Simplified 98.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -280000:\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;y \leq 1.75 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+142}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))))
   (if (<= y 1.75e+21) t_0 (if (<= y 1.9e+142) (/ (* x (- y)) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (y <= 1.75e+21) {
		tmp = t_0;
	} else if (y <= 1.9e+142) {
		tmp = (x * -y) / 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 (y <= 1.75d+21) then
        tmp = t_0
    else if (y <= 1.9d+142) then
        tmp = (x * -y) / 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 (y <= 1.75e+21) {
		tmp = t_0;
	} else if (y <= 1.9e+142) {
		tmp = (x * -y) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x / z)
	tmp = 0
	if y <= 1.75e+21:
		tmp = t_0
	elif y <= 1.9e+142:
		tmp = (x * -y) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x / z))
	tmp = 0.0
	if (y <= 1.75e+21)
		tmp = t_0;
	elseif (y <= 1.9e+142)
		tmp = Float64(Float64(x * Float64(-y)) / 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 (y <= 1.75e+21)
		tmp = t_0;
	elseif (y <= 1.9e+142)
		tmp = (x * -y) / 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[y, 1.75e+21], t$95$0, If[LessEqual[y, 1.9e+142], N[(N[(x * (-y)), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.75e21 or 1.89999999999999995e142 < y

   1. Initial program 82.1%

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

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. div-sub N/A

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

      15. unsub-neg N/A

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

      16. mul-1-neg N/A

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

   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 82.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 82.1

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

   11. Simplified 82.1%

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

   if 1.75e21 < y < 1.89999999999999995e142

   1. Initial program 88.8%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 88.8

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

   5. Simplified 88.8%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 67.9

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

   8. Simplified 67.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+21}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+142}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;-x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))))
   (if (<= y 1.4e+21) t_0 (if (<= y 1.6e+143) (- (* x (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (y <= 1.4e+21) {
		tmp = t_0;
	} else if (y <= 1.6e+143) {
		tmp = -(x * (y / 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 (y <= 1.4d+21) then
        tmp = t_0
    else if (y <= 1.6d+143) then
        tmp = -(x * (y / 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 (y <= 1.4e+21) {
		tmp = t_0;
	} else if (y <= 1.6e+143) {
		tmp = -(x * (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x / z)
	tmp = 0
	if y <= 1.4e+21:
		tmp = t_0
	elif y <= 1.6e+143:
		tmp = -(x * (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x / z))
	tmp = 0.0
	if (y <= 1.4e+21)
		tmp = t_0;
	elseif (y <= 1.6e+143)
		tmp = Float64(-Float64(x * Float64(y / 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 (y <= 1.4e+21)
		tmp = t_0;
	elseif (y <= 1.6e+143)
		tmp = -(x * (y / 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[y, 1.4e+21], t$95$0, If[LessEqual[y, 1.6e+143], (-N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4e21 or 1.60000000000000008e143 < y

   1. Initial program 82.1%

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

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. div-sub N/A

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

      15. unsub-neg N/A

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

      16. mul-1-neg N/A

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

   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 82.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 82.1

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

   11. Simplified 82.1%

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

   if 1.4e21 < y < 1.60000000000000008e143

   1. Initial program 88.8%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 92.5

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

   5. Simplified 92.5%

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

   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 67.7

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

   8. Simplified 67.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+21}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;-x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. 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 82.7%

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

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

   11. +-commutative N/A

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

   12. mul-1-neg N/A

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

   13. unsub-neg N/A

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

   14. div-sub N/A

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

   15. unsub-neg N/A

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

   16. mul-1-neg N/A

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

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 76.5%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-+.f64 N/A

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

    3. lower-/.f64 76.5

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

11. Simplified 76.5%

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

12. Final simplification 76.5%

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

Alternative 7: 40.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 32.4

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

5. Simplified 32.4%

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

Developer Target 1: 93.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024215 
(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))