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

Percentage Accurate: 88.4% → 100.0%

Time: 9.3s

Alternatives: 9

Speedup: 1.1×

• 20.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

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

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. div-sub N/A

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

   14. unsub-neg N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

5. Simplified 99.9%

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

Alternative 2: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq -2 \cdot 10^{+225}:\\ \;\;\;\;-y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (+ x (* y (- z x))) z) -2e+225) (- (* y (/ x z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * (z - x))) / z) <= -2e+225) {
		tmp = -(y * (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x + (y * (z - x))) / z) <= (-2d+225)) then
        tmp = -(y * (x / z))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * (z - x))) / z) <= -2e+225) {
		tmp = -(y * (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x + (y * (z - x))) / z) <= -2e+225:
		tmp = -(y * (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + Float64(y * Float64(z - x))) / z) <= -2e+225)
		tmp = Float64(-Float64(y * Float64(x / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x + (y * (z - x))) / z) <= -2e+225)
		tmp = -(y * (x / z));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -2e+225], (-N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z) < -1.99999999999999986e225

   1. Initial program 75.7%

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

   3. Taylor expanded in y around inf

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

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

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

      2. --lowering--.f64 59.5

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

   5. Simplified 59.5%

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

   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. neg-lowering-neg.f64 52.3

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

   8. Simplified 52.3%

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

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

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

      7. *-lft-identity N/A

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

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

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

      9. neg-lowering-neg.f64 62.7

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

   10. Applied egg-rr 62.7%

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

   if -1.99999999999999986e225 < (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z)

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Simplified 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 87.2%

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

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

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

      2. *-lft-identity N/A

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

      3. /-lowering-/.f64 87.2

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

   10. Applied egg-rr 87.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq -2 \cdot 10^{+225}:\\ \;\;\;\;-y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq -2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \left(-\frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (+ x (* y (- z x))) z) -2e+225) (* x (- (/ y z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * (z - x))) / z) <= -2e+225) {
		tmp = x * -(y / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x + (y * (z - x))) / z) <= (-2d+225)) then
        tmp = x * -(y / z)
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * (z - x))) / z) <= -2e+225) {
		tmp = x * -(y / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x + (y * (z - x))) / z) <= -2e+225:
		tmp = x * -(y / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + Float64(y * Float64(z - x))) / z) <= -2e+225)
		tmp = Float64(x * Float64(-Float64(y / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x + (y * (z - x))) / z) <= -2e+225)
		tmp = x * -(y / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -2e+225], N[(x * (-N[(y / z), $MachinePrecision])), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z) < -1.99999999999999986e225

   1. Initial program 75.7%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. --lowering--.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. *-lowering-*.f64 68.5

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

   5. Simplified 68.5%

      \[\leadsto \frac{\color{blue}{x - y \cdot x}}{z} \]
   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. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. --lowering--.f64 73.7

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

   7. Applied egg-rr 73.7%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 57.5

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

   10. Simplified 57.5%

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

   if -1.99999999999999986e225 < (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z)

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Simplified 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 87.2%

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

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

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

      2. *-lft-identity N/A

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

      3. /-lowering-/.f64 87.2

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

   10. Applied egg-rr 87.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq -2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \left(-\frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ x z) y)))
   (if (<= y -1.0) t_0 (if (<= y 2.2e-10) (+ 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 <= -1.0) {
		tmp = t_0;
	} else if (y <= 2.2e-10) {
		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 <= -1.0)
		tmp = t_0;
	elseif (y <= 2.2e-10)
		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, -1.0], t$95$0, If[LessEqual[y, 2.2e-10], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.1999999999999999e-10 < y

   1. Initial program 75.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. 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. neg-lowering-neg.f64 98.8

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

   8. Simplified 98.8%

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

   if -1 < y < 2.1999999999999999e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 99.5%

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

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

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

      2. *-lft-identity N/A

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

      3. /-lowering-/.f64 99.5

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.0)
   (- y (/ (* y x) z))
   (if (<= y 2.2e-10) (+ y (/ x z)) (* (- z x) (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.0) {
		tmp = y - ((y * x) / z);
	} else if (y <= 2.2e-10) {
		tmp = y + (x / z);
	} else {
		tmp = (z - x) * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = y - ((y * x) / z)
    else if (y <= 2.2d-10) then
        tmp = y + (x / z)
    else
        tmp = (z - x) * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.0) {
		tmp = y - ((y * x) / z);
	} else if (y <= 2.2e-10) {
		tmp = y + (x / z);
	} else {
		tmp = (z - x) * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.0:
		tmp = y - ((y * x) / z)
	elif y <= 2.2e-10:
		tmp = y + (x / z)
	else:
		tmp = (z - x) * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(y - Float64(Float64(y * x) / z));
	elseif (y <= 2.2e-10)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(Float64(z - x) * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = y - ((y * x) / z);
	elseif (y <= 2.2e-10)
		tmp = y + (x / z);
	else
		tmp = (z - x) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.0], N[(y - N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-10], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(z - x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 78.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. --lowering--.f64 N/A

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

      11. /-lowering-/.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. *-lowering-*.f64 91.8

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

   5. Simplified 91.8%

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

   if -1 < y < 2.1999999999999999e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 99.5%

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

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

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

      2. *-lft-identity N/A

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

      3. /-lowering-/.f64 99.5

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

   10. Applied egg-rr 99.5%

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

   if 2.1999999999999999e-10 < y

   1. Initial program 72.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 71.4

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

   5. Simplified 71.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 93.1

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

   7. Applied egg-rr 93.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z - x\right) \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z x) (/ y z))))
   (if (<= y -1.0) t_0 (if (<= y 2.2e-10) (+ y (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z - x) * (y / z);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 2.2e-10) {
		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 = (z - x) * (y / z)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 2.2d-10) 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 = (z - x) * (y / z);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 2.2e-10) {
		tmp = y + (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z - x) * (y / z)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 2.2e-10:
		tmp = y + (x / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z - x) * Float64(y / z))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 2.2e-10)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z - x) * (y / z);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 2.2e-10)
		tmp = y + (x / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z - x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 2.2e-10], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.1999999999999999e-10 < y

   1. Initial program 75.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 74.0

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

   5. Simplified 74.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 90.9

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

   7. Applied egg-rr 90.9%

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

   if -1 < y < 2.1999999999999999e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 99.5%

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

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

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

      2. *-lft-identity N/A

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

      3. /-lowering-/.f64 99.5

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-26}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.2e-26) y (if (<= y 2.25e-71) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.2e-26) {
		tmp = y;
	} else if (y <= 2.25e-71) {
		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 <= (-5.2d-26)) then
        tmp = y
    else if (y <= 2.25d-71) 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 <= -5.2e-26) {
		tmp = y;
	} else if (y <= 2.25e-71) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.2e-26:
		tmp = y
	elif y <= 2.25e-71:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.2e-26)
		tmp = y;
	elseif (y <= 2.25e-71)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.2e-26)
		tmp = y;
	elseif (y <= 2.25e-71)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.2e-26], y, If[LessEqual[y, 2.25e-71], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2000000000000002e-26 or 2.2500000000000001e-71 < y

   1. Initial program 78.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 54.9%

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

   if -5.2000000000000002e-26 < y < 2.2500000000000001e-71

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 76.8%

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

Alternative 8: 78.9% 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 88.2%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

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

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. div-sub N/A

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

   14. unsub-neg N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

5. Simplified 99.9%

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

6. Taylor expanded in y around 0

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

8. Simplified 78.9%

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

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

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

   2. *-lft-identity N/A

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

   3. /-lowering-/.f64 78.9

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

10. Applied egg-rr 78.9%

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

11. Final simplification 78.9%

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

Alternative 9: 41.6% 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 88.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 41.4%

   \[\leadsto \color{blue}{y} \]
6. 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 2024199 
(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))