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

Percentage Accurate: 88.8% → 99.9%

Time: 10.3s

Alternatives: 10

Speedup: 1.1×

• 20.7% 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: 88.8% 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 89.5%

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

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

Alternative 2: 77.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+197}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+14}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+263}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y (/ x z)))))
   (if (<= y -2.1e+197)
     t_0
     (if (<= y 1.8e+14) (+ y (/ x z)) (if (<= y 5.2e+263) t_0 y)))))

C

double code(double x, double y, double z) {
	double t_0 = -(y * (x / z));
	double tmp;
	if (y <= -2.1e+197) {
		tmp = t_0;
	} else if (y <= 1.8e+14) {
		tmp = y + (x / z);
	} else if (y <= 5.2e+263) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -(y * (x / z))
    if (y <= (-2.1d+197)) then
        tmp = t_0
    else if (y <= 1.8d+14) then
        tmp = y + (x / z)
    else if (y <= 5.2d+263) then
        tmp = t_0
    else
        tmp = y
    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 <= -2.1e+197) {
		tmp = t_0;
	} else if (y <= 1.8e+14) {
		tmp = y + (x / z);
	} else if (y <= 5.2e+263) {
		tmp = t_0;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(y * (x / z))
	tmp = 0
	if y <= -2.1e+197:
		tmp = t_0
	elif y <= 1.8e+14:
		tmp = y + (x / z)
	elif y <= 5.2e+263:
		tmp = t_0
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(y * Float64(x / z)))
	tmp = 0.0
	if (y <= -2.1e+197)
		tmp = t_0;
	elseif (y <= 1.8e+14)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 5.2e+263)
		tmp = t_0;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(y * (x / z));
	tmp = 0.0;
	if (y <= -2.1e+197)
		tmp = t_0;
	elseif (y <= 1.8e+14)
		tmp = y + (x / z);
	elseif (y <= 5.2e+263)
		tmp = t_0;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[y, -2.1e+197], t$95$0, If[LessEqual[y, 1.8e+14], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e+263], t$95$0, y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.10000000000000006e197 or 1.8e14 < y < 5.2000000000000004e263

   1. Initial program 83.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 94.9

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 62.9

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

   8. Applied rewrites 62.9%

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

      1. distribute-rgt-neg-out N/A

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

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

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 66.5

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

   10. Applied rewrites 66.5%

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

   if -2.10000000000000006e197 < y < 1.8e14

   1. Initial program 94.8%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 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. Applied rewrites 90.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{x}{z}, y\right) \]
   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 90.7

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

      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 90.7

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

   10. Applied rewrites 90.7%

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

   if 5.2000000000000004e263 < y

   1. Initial program 45.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 26.2

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

   5. Applied rewrites 26.2%

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

      1. associate-/l* N/A

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

      2. *-inverses N/A

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

      3. *-rgt-identity 81.0

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

   7. Applied rewrites 81.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+197}:\\ \;\;\;\;-y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+14}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+263}:\\ \;\;\;\;-y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (* y (/ x z)))))
   (if (<= y -5e+20) t_0 (if (<= y 5e+16) (/ (fma (- z x) 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 <= -5e+20) {
		tmp = t_0;
	} else if (y <= 5e+16) {
		tmp = fma((z - x), y, x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(y * Float64(x / z)))
	tmp = 0.0
	if (y <= -5e+20)
		tmp = t_0;
	elseif (y <= 5e+16)
		tmp = Float64(fma(Float64(z - x), y, x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5e+20], t$95$0, If[LessEqual[y, 5e+16], N[(N[(N[(z - x), $MachinePrecision] * y + x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5e20 or 5e16 < y

   1. Initial program 77.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 95.2

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

   5. Applied rewrites 95.2%

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

      1. associate-/l* N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -5e20 < y < 5e16

   1. Initial program 99.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+20}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -95000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (* y (/ x z)))))
   (if (<= y -95000.0) 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 <= -95000.0) {
		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 <= (-95000.0d0)) 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 <= -95000.0) {
		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 <= -95000.0:
		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(y * Float64(x / z)))
	tmp = 0.0
	if (y <= -95000.0)
		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 <= -95000.0)
		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[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -95000.0], 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 < -95000 or 1 < y

   1. Initial program 79.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. sub-neg N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

      1. associate-/l* N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if -95000 < 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. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.0%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity 98.0

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

      5. lift-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-lft-identity 98.0

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

   10. Applied rewrites 98.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -95000:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - x \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -95000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (* x (/ y z)))))
   (if (<= y -95000.0) t_0 (if (<= y 1.0) (+ y (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y - (x * (y / z));
	double tmp;
	if (y <= -95000.0) {
		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 - (x * (y / z))
    if (y <= (-95000.0d0)) 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 - (x * (y / z));
	double tmp;
	if (y <= -95000.0) {
		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 - (x * (y / z))
	tmp = 0
	if y <= -95000.0:
		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(x * Float64(y / z)))
	tmp = 0.0
	if (y <= -95000.0)
		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 - (x * (y / z));
	tmp = 0.0;
	if (y <= -95000.0)
		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[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -95000.0], 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 < -95000 or 1 < y

   1. Initial program 79.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. sub-neg N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 93.1

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

   7. Applied rewrites 93.1%

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

   if -95000 < 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. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.0%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity 98.0

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

      5. lift-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-lft-identity 98.0

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

   10. Applied rewrites 98.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -95000:\\ \;\;\;\;y - x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-94}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))))
   (if (<= z -4.8e-20) t_0 (if (<= z 1.75e-94) (* (- 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 <= -4.8e-20) {
		tmp = t_0;
	} else if (z <= 1.75e-94) {
		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 <= (-4.8d-20)) then
        tmp = t_0
    else if (z <= 1.75d-94) 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 <= -4.8e-20) {
		tmp = t_0;
	} else if (z <= 1.75e-94) {
		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 <= -4.8e-20:
		tmp = t_0
	elif z <= 1.75e-94:
		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 <= -4.8e-20)
		tmp = t_0;
	elseif (z <= 1.75e-94)
		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 <= -4.8e-20)
		tmp = t_0;
	elseif (z <= 1.75e-94)
		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, -4.8e-20], t$95$0, If[LessEqual[z, 1.75e-94], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.79999999999999986e-20 or 1.74999999999999999e-94 < z

   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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 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. Applied rewrites 84.4%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity 84.4

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

      5. lift-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-lft-identity 84.4

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

   10. Applied rewrites 84.4%

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

   if -4.79999999999999986e-20 < z < 1.74999999999999999e-94

   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 93.7

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

   5. Applied rewrites 93.7%

      \[\leadsto \frac{\color{blue}{x - y \cdot x}}{z} \]
   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 93.8

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

   7. Applied rewrites 93.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-20}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-94}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3950000000000:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3950000000000.0) y (if (<= y 8.5e-79) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3950000000000.0) {
		tmp = y;
	} else if (y <= 8.5e-79) {
		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 <= (-3950000000000.0d0)) then
        tmp = y
    else if (y <= 8.5d-79) 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 <= -3950000000000.0) {
		tmp = y;
	} else if (y <= 8.5e-79) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3950000000000.0:
		tmp = y
	elif y <= 8.5e-79:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3950000000000.0)
		tmp = y;
	elseif (y <= 8.5e-79)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3950000000000.0)
		tmp = y;
	elseif (y <= 8.5e-79)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3950000000000.0], y, If[LessEqual[y, 8.5e-79], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -3.95e12 or 8.50000000000000029e-79 < y

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 32.4

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

   5. Applied rewrites 32.4%

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

      1. associate-/l* N/A

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

      2. *-inverses N/A

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

      3. *-rgt-identity 47.2

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

   7. Applied rewrites 47.2%

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

   if -3.95e12 < y < 8.50000000000000029e-79

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

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

Alternative 8: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{+104}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.8e+104) (+ y (/ x z)) (* z (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.8e+104) {
		tmp = y + (x / z);
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5.8d+104) then
        tmp = y + (x / z)
    else
        tmp = z * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.8e+104) {
		tmp = y + (x / z);
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.8e+104:
		tmp = y + (x / z)
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.8e+104)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.8e+104)
		tmp = y + (x / z);
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.8e+104], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.7999999999999997e104

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 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. Applied rewrites 79.0%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity 79.0

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

      5. lift-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-lft-identity 79.0

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

   10. Applied rewrites 79.0%

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

   if 5.7999999999999997e104 < y

   1. Initial program 72.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 26.8

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

   5. Applied rewrites 26.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 58.6

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

   7. Applied rewrites 58.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{+104}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.3% 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 89.5%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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. Applied rewrites 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. Applied rewrites 74.2%

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

   1. lift-/.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lower-fma.f64 N/A

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

   4. *-lft-identity 74.2

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

   5. lift-fma.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. *-lft-identity 74.2

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

10. Applied rewrites 74.2%

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

11. Final simplification 74.2%

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

Alternative 10: 41.0% 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 89.5%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 27.4

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

5. Applied rewrites 27.4%

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

   1. associate-/l* N/A

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

   2. *-inverses N/A

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

   3. *-rgt-identity 35.6

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

7. Applied rewrites 35.6%

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

Developer Target 1: 94.2% 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 2024214 
(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))