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

Percentage Accurate: 89.0% → 99.9%

Time: 6.5s

Alternatives: 10

Speedup: 1.0×

• 22.1% 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: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + (-1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.9%

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

2. Taylor expanded in x around -inf 96.6%

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

   1. mul-1-neg 96.6%

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

   2. unsub-neg 96.6%

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

   3. associate-/l* 97.1%

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

   4. associate-/r/ 99.9%

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

   5. sub-neg 99.9%

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

   6. metadata-eval 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 76.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-y}{z}\\ t_1 := y + \frac{x}{z}\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{+272}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7300000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ (- y) z))) (t_1 (+ y (/ x z))))
   (if (<= y -2.05e+272)
     t_0
     (if (<= y -5.8e+87)
       t_1
       (if (<= y -3e+39)
         t_0
         (if (<= y 7300000.0) t_1 (if (<= y 9.2e+120) t_0 (* z (/ y z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (-y / z);
	double t_1 = y + (x / z);
	double tmp;
	if (y <= -2.05e+272) {
		tmp = t_0;
	} else if (y <= -5.8e+87) {
		tmp = t_1;
	} else if (y <= -3e+39) {
		tmp = t_0;
	} else if (y <= 7300000.0) {
		tmp = t_1;
	} else if (y <= 9.2e+120) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (-y / z)
    t_1 = y + (x / z)
    if (y <= (-2.05d+272)) then
        tmp = t_0
    else if (y <= (-5.8d+87)) then
        tmp = t_1
    else if (y <= (-3d+39)) then
        tmp = t_0
    else if (y <= 7300000.0d0) then
        tmp = t_1
    else if (y <= 9.2d+120) then
        tmp = t_0
    else
        tmp = z * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (-y / z);
	double t_1 = y + (x / z);
	double tmp;
	if (y <= -2.05e+272) {
		tmp = t_0;
	} else if (y <= -5.8e+87) {
		tmp = t_1;
	} else if (y <= -3e+39) {
		tmp = t_0;
	} else if (y <= 7300000.0) {
		tmp = t_1;
	} else if (y <= 9.2e+120) {
		tmp = t_0;
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (-y / z)
	t_1 = y + (x / z)
	tmp = 0
	if y <= -2.05e+272:
		tmp = t_0
	elif y <= -5.8e+87:
		tmp = t_1
	elif y <= -3e+39:
		tmp = t_0
	elif y <= 7300000.0:
		tmp = t_1
	elif y <= 9.2e+120:
		tmp = t_0
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(-y) / z))
	t_1 = Float64(y + Float64(x / z))
	tmp = 0.0
	if (y <= -2.05e+272)
		tmp = t_0;
	elseif (y <= -5.8e+87)
		tmp = t_1;
	elseif (y <= -3e+39)
		tmp = t_0;
	elseif (y <= 7300000.0)
		tmp = t_1;
	elseif (y <= 9.2e+120)
		tmp = t_0;
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (-y / z);
	t_1 = y + (x / z);
	tmp = 0.0;
	if (y <= -2.05e+272)
		tmp = t_0;
	elseif (y <= -5.8e+87)
		tmp = t_1;
	elseif (y <= -3e+39)
		tmp = t_0;
	elseif (y <= 7300000.0)
		tmp = t_1;
	elseif (y <= 9.2e+120)
		tmp = t_0;
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.05e+272], t$95$0, If[LessEqual[y, -5.8e+87], t$95$1, If[LessEqual[y, -3e+39], t$95$0, If[LessEqual[y, 7300000.0], t$95$1, If[LessEqual[y, 9.2e+120], t$95$0, N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.04999999999999989e272 or -5.7999999999999996e87 < y < -3e39 or 7.3e6 < y < 9.1999999999999997e120

   1. Initial program 81.9%

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

   2. Taylor expanded in y around inf 81.9%

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

      1. associate-/l* 99.8%

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

      2. associate-/r/ 97.8%

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

   4. Simplified 97.8%

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

   5. Taylor expanded in z around 0 66.4%

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

      1. associate-*r/ 71.9%

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

      2. associate-*r* 71.9%

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

      3. neg-mul-1 71.9%

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

      4. *-commutative 71.9%

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

   7. Simplified 71.9%

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

   if -2.04999999999999989e272 < y < -5.7999999999999996e87 or -3e39 < y < 7.3e6

   1. Initial program 94.8%

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

   2. Taylor expanded in x around -inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. associate-/l* 98.5%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 91.3%

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

      1. mul-1-neg 91.3%

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

      2. distribute-frac-neg 91.3%

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

   7. Simplified 91.3%

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

   8. Taylor expanded in y around 0 91.3%

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

      1. +-commutative 91.3%

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

   10. Simplified 91.3%

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

   if 9.1999999999999997e120 < y

   1. Initial program 73.7%

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

   2. Taylor expanded in x around 0 34.4%

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

      1. associate-/l* 56.2%

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

      2. associate-/r/ 67.0%

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

   4. Applied egg-rr 67.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+272}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+87}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;y \leq 7300000:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+120}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]

Alternative 3: 77.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\frac{-z}{x}}\\ t_1 := y + \frac{x}{z}\\ \mathbf{if}\;y \leq -2.15 \cdot 10^{+272}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1400000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (/ (- z) x))) (t_1 (+ y (/ x z))))
   (if (<= y -2.15e+272)
     (* x (/ (- y) z))
     (if (<= y -1.5e+89)
       t_1
       (if (<= y -3.1e+39)
         t_0
         (if (<= y 1400000000000.0)
           t_1
           (if (<= y 1.56e+121) t_0 (* z (/ y z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (-z / x);
	double t_1 = y + (x / z);
	double tmp;
	if (y <= -2.15e+272) {
		tmp = x * (-y / z);
	} else if (y <= -1.5e+89) {
		tmp = t_1;
	} else if (y <= -3.1e+39) {
		tmp = t_0;
	} else if (y <= 1400000000000.0) {
		tmp = t_1;
	} else if (y <= 1.56e+121) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / (-z / x)
    t_1 = y + (x / z)
    if (y <= (-2.15d+272)) then
        tmp = x * (-y / z)
    else if (y <= (-1.5d+89)) then
        tmp = t_1
    else if (y <= (-3.1d+39)) then
        tmp = t_0
    else if (y <= 1400000000000.0d0) then
        tmp = t_1
    else if (y <= 1.56d+121) then
        tmp = t_0
    else
        tmp = z * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (-z / x);
	double t_1 = y + (x / z);
	double tmp;
	if (y <= -2.15e+272) {
		tmp = x * (-y / z);
	} else if (y <= -1.5e+89) {
		tmp = t_1;
	} else if (y <= -3.1e+39) {
		tmp = t_0;
	} else if (y <= 1400000000000.0) {
		tmp = t_1;
	} else if (y <= 1.56e+121) {
		tmp = t_0;
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (-z / x)
	t_1 = y + (x / z)
	tmp = 0
	if y <= -2.15e+272:
		tmp = x * (-y / z)
	elif y <= -1.5e+89:
		tmp = t_1
	elif y <= -3.1e+39:
		tmp = t_0
	elif y <= 1400000000000.0:
		tmp = t_1
	elif y <= 1.56e+121:
		tmp = t_0
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(Float64(-z) / x))
	t_1 = Float64(y + Float64(x / z))
	tmp = 0.0
	if (y <= -2.15e+272)
		tmp = Float64(x * Float64(Float64(-y) / z));
	elseif (y <= -1.5e+89)
		tmp = t_1;
	elseif (y <= -3.1e+39)
		tmp = t_0;
	elseif (y <= 1400000000000.0)
		tmp = t_1;
	elseif (y <= 1.56e+121)
		tmp = t_0;
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (-z / x);
	t_1 = y + (x / z);
	tmp = 0.0;
	if (y <= -2.15e+272)
		tmp = x * (-y / z);
	elseif (y <= -1.5e+89)
		tmp = t_1;
	elseif (y <= -3.1e+39)
		tmp = t_0;
	elseif (y <= 1400000000000.0)
		tmp = t_1;
	elseif (y <= 1.56e+121)
		tmp = t_0;
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[((-z) / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.15e+272], N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.5e+89], t$95$1, If[LessEqual[y, -3.1e+39], t$95$0, If[LessEqual[y, 1400000000000.0], t$95$1, If[LessEqual[y, 1.56e+121], t$95$0, N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.15000000000000002e272

   1. Initial program 88.6%

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

   2. Taylor expanded in y around inf 88.6%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 87.9%

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

      1. associate-*r/ 87.9%

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

      2. associate-*r* 87.9%

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

      3. neg-mul-1 87.9%

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

      4. *-commutative 87.9%

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

   7. Simplified 87.9%

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

   if -2.15000000000000002e272 < y < -1.50000000000000006e89 or -3.1000000000000003e39 < y < 1.4e12

   1. Initial program 94.8%

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

   2. Taylor expanded in x around -inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. associate-/l* 98.5%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 91.3%

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

      1. mul-1-neg 91.3%

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

      2. distribute-frac-neg 91.3%

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

   7. Simplified 91.3%

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

   8. Taylor expanded in y around 0 91.3%

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

      1. +-commutative 91.3%

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

   10. Simplified 91.3%

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

   if -1.50000000000000006e89 < y < -3.1000000000000003e39 or 1.4e12 < y < 1.5599999999999999e121

   1. Initial program 80.6%

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

   2. Taylor expanded in y around inf 80.6%

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

      1. associate-/l* 99.8%

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

      2. associate-/r/ 97.4%

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

   4. Simplified 97.4%

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

      1. associate-*l/ 80.6%

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

      2. associate-/l* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around 0 71.0%

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

      1. associate-*r/ 71.0%

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

      2. neg-mul-1 71.0%

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

   9. Simplified 71.0%

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

   if 1.5599999999999999e121 < y

   1. Initial program 73.7%

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

   2. Taylor expanded in x around 0 34.4%

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

      1. associate-/l* 56.2%

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

      2. associate-/r/ 67.0%

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

   4. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+272}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{+89}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{elif}\;y \leq 1400000000000:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+121}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]

Alternative 4: 60.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -5 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{-85}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ y z))))
   (if (<= y -5e-11)
     t_0
     (if (<= y -2.25e-74)
       (/ x z)
       (if (<= y -1.26e-85) y (if (<= y 2.2e-26) (/ x z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y / z);
	double tmp;
	if (y <= -5e-11) {
		tmp = t_0;
	} else if (y <= -2.25e-74) {
		tmp = x / z;
	} else if (y <= -1.26e-85) {
		tmp = y;
	} else if (y <= 2.2e-26) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (y / z)
    if (y <= (-5d-11)) then
        tmp = t_0
    else if (y <= (-2.25d-74)) then
        tmp = x / z
    else if (y <= (-1.26d-85)) then
        tmp = y
    else if (y <= 2.2d-26) then
        tmp = x / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y / z);
	double tmp;
	if (y <= -5e-11) {
		tmp = t_0;
	} else if (y <= -2.25e-74) {
		tmp = x / z;
	} else if (y <= -1.26e-85) {
		tmp = y;
	} else if (y <= 2.2e-26) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y / z)
	tmp = 0
	if y <= -5e-11:
		tmp = t_0
	elif y <= -2.25e-74:
		tmp = x / z
	elif y <= -1.26e-85:
		tmp = y
	elif y <= 2.2e-26:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y / z))
	tmp = 0.0
	if (y <= -5e-11)
		tmp = t_0;
	elseif (y <= -2.25e-74)
		tmp = Float64(x / z);
	elseif (y <= -1.26e-85)
		tmp = y;
	elseif (y <= 2.2e-26)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y / z);
	tmp = 0.0;
	if (y <= -5e-11)
		tmp = t_0;
	elseif (y <= -2.25e-74)
		tmp = x / z;
	elseif (y <= -1.26e-85)
		tmp = y;
	elseif (y <= 2.2e-26)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5e-11], t$95$0, If[LessEqual[y, -2.25e-74], N[(x / z), $MachinePrecision], If[LessEqual[y, -1.26e-85], y, If[LessEqual[y, 2.2e-26], N[(x / z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.00000000000000018e-11 or 2.2000000000000001e-26 < y

   1. Initial program 80.0%

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

   2. Taylor expanded in x around 0 31.5%

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

      1. associate-/l* 46.7%

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

      2. associate-/r/ 55.5%

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

   4. Applied egg-rr 55.5%

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

   if -5.00000000000000018e-11 < y < -2.25e-74 or -1.26e-85 < y < 2.2000000000000001e-26

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 75.7%

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

   if -2.25e-74 < y < -1.26e-85

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-11}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{-85}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]

Alternative 5: 58.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-18}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-74} \lor \neg \left(y \leq -1.25 \cdot 10^{-85}\right) \land y \leq 9.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e-18)
   y
   (if (or (<= y -2.2e-74) (and (not (<= y -1.25e-85)) (<= y 9.5e-27)))
     (/ x z)
     y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e-18) {
		tmp = y;
	} else if ((y <= -2.2e-74) || (!(y <= -1.25e-85) && (y <= 9.5e-27))) {
		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 <= (-2.2d-18)) then
        tmp = y
    else if ((y <= (-2.2d-74)) .or. (.not. (y <= (-1.25d-85))) .and. (y <= 9.5d-27)) 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 <= -2.2e-18) {
		tmp = y;
	} else if ((y <= -2.2e-74) || (!(y <= -1.25e-85) && (y <= 9.5e-27))) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.2e-18:
		tmp = y
	elif (y <= -2.2e-74) or (not (y <= -1.25e-85) and (y <= 9.5e-27)):
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e-18)
		tmp = y;
	elseif ((y <= -2.2e-74) || (!(y <= -1.25e-85) && (y <= 9.5e-27)))
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.2e-18)
		tmp = y;
	elseif ((y <= -2.2e-74) || (~((y <= -1.25e-85)) && (y <= 9.5e-27)))
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e-18], y, If[Or[LessEqual[y, -2.2e-74], And[N[Not[LessEqual[y, -1.25e-85]], $MachinePrecision], LessEqual[y, 9.5e-27]]], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1999999999999998e-18 or -2.2000000000000001e-74 < y < -1.25e-85 or 9.50000000000000037e-27 < y

   1. Initial program 80.6%

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

   2. Taylor expanded in x around 0 48.6%

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

   if -2.1999999999999998e-18 < y < -2.2000000000000001e-74 or -1.25e-85 < y < 9.50000000000000037e-27

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 75.7%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-18}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-74} \lor \neg \left(y \leq -1.25 \cdot 10^{-85}\right) \land y \leq 9.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10500000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -10500000000.0) (not (<= y 1.0)))
   (* (- z x) (/ y z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -10500000000.0) || !(y <= 1.0)) {
		tmp = (z - 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 ((y <= (-10500000000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = (z - 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 ((y <= -10500000000.0) || !(y <= 1.0)) {
		tmp = (z - x) * (y / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -10500000000.0) or not (y <= 1.0):
		tmp = (z - x) * (y / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -10500000000.0) || !(y <= 1.0))
		tmp = Float64(Float64(z - x) * 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 ((y <= -10500000000.0) || ~((y <= 1.0)))
		tmp = (z - x) * (y / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -10500000000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[(z - x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e10 or 1 < y

   1. Initial program 78.4%

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

   2. Taylor expanded in y around inf 78.0%

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

      1. associate-/l* 99.4%

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

      2. associate-/r/ 92.9%

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

   4. Simplified 92.9%

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

   if -1.05e10 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10500000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 7: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10500000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -10500000000.0) (not (<= y 1.0)))
   (/ y (/ z (- z x)))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -10500000000.0) || !(y <= 1.0)) {
		tmp = y / (z / (z - x));
	} 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 ((y <= (-10500000000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y / (z / (z - x))
    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 ((y <= -10500000000.0) || !(y <= 1.0)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -10500000000.0) or not (y <= 1.0):
		tmp = y / (z / (z - x))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -10500000000.0) || !(y <= 1.0))
		tmp = Float64(y / Float64(z / Float64(z - x)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -10500000000.0) || ~((y <= 1.0)))
		tmp = y / (z / (z - x));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -10500000000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e10 or 1 < y

   1. Initial program 78.4%

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

   2. Taylor expanded in y around inf 78.0%

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

      1. associate-/l* 99.4%

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

      2. associate-/r/ 92.9%

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

   4. Simplified 92.9%

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

      1. associate-*l/ 78.0%

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

      2. associate-/l* 99.4%

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

   6. Applied egg-rr 99.4%

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

   if -1.05e10 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10500000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 8: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 11500:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 11500.0) (+ y (/ x z)) (* z (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 11500.0) {
		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 <= 11500.0d0) 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 <= 11500.0) {
		tmp = y + (x / z);
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 11500.0:
		tmp = y + (x / z)
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 11500.0)
		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 <= 11500.0)
		tmp = y + (x / z);
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 11500.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 11500

   1. Initial program 93.4%

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

   2. Taylor expanded in x around -inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. associate-/l* 98.7%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 84.9%

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

      1. mul-1-neg 84.9%

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

      2. distribute-frac-neg 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in y around 0 84.9%

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

      1. +-commutative 84.9%

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

   10. Simplified 84.9%

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

   if 11500 < y

   1. Initial program 76.6%

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

   2. Taylor expanded in x around 0 29.4%

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

      1. associate-/l* 47.1%

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

      2. associate-/r/ 56.6%

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

   4. Applied egg-rr 56.6%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 11500:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]

Alternative 9: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.0) (+ y (/ x z)) (/ z (/ z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = z / (z / 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 <= 1.0d0) then
        tmp = y + (x / z)
    else
        tmp = z / (z / y)
    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 + (x / z);
	} else {
		tmp = z / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = z / (z / y)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = z / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 93.3%

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

   2. Taylor expanded in x around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. associate-/l* 98.7%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 85.3%

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

      1. mul-1-neg 85.3%

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

      2. distribute-frac-neg 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in y around 0 85.3%

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

      1. +-commutative 85.3%

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

   10. Simplified 85.3%

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

   if 1 < y

   1. Initial program 77.3%

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

   2. Taylor expanded in x around 0 30.0%

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

      1. associate-/l* 47.2%

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

      2. associate-/r/ 56.4%

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

   4. Applied egg-rr 56.4%

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

      1. *-commutative 56.4%

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

      2. clear-num 56.3%

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

      3. un-div-inv 56.4%

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

   6. Applied egg-rr 56.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \end{array} \]

Alternative 10: 39.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 88.9%

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

2. Taylor expanded in x around 0 39.0%

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

3. Final simplification 39.0%

   \[\leadsto y \]

Developer target: 93.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023320 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))