Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Percentage Accurate: 87.6% → 97.4%

Time: 5.9s

Alternatives: 10

Speedup: 1.0×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / 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) + 1.0d0)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / 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) + 1.0d0)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;\frac{x \cdot t_0}{z} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- y z) 1.0)))
   (if (<= (/ (* x t_0) z) 5e-93)
     (/ x (/ z t_0))
     (* (/ x z) (- (+ y 1.0) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) + 1.0;
	double tmp;
	if (((x * t_0) / z) <= 5e-93) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x / z) * ((y + 1.0) - 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) :: tmp
    t_0 = (y - z) + 1.0d0
    if (((x * t_0) / z) <= 5d-93) then
        tmp = x / (z / t_0)
    else
        tmp = (x / z) * ((y + 1.0d0) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - z) + 1.0;
	double tmp;
	if (((x * t_0) / z) <= 5e-93) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x / z) * ((y + 1.0) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - z) + 1.0
	tmp = 0
	if ((x * t_0) / z) <= 5e-93:
		tmp = x / (z / t_0)
	else:
		tmp = (x / z) * ((y + 1.0) - z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) + 1.0)
	tmp = 0.0
	if (Float64(Float64(x * t_0) / z) <= 5e-93)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y + 1.0) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - z) + 1.0;
	tmp = 0.0;
	if (((x * t_0) / z) <= 5e-93)
		tmp = x / (z / t_0);
	else
		tmp = (x / z) * ((y + 1.0) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], 5e-93], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 4.99999999999999994e-93

   1. Initial program 89.9%

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

      1. associate-/l* 99.2%

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

   3. Simplified 99.2%

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

   if 4.99999999999999994e-93 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 83.8%

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

      1. associate-/l* 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in x around 0 83.8%

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

      1. associate--l+ 83.8%

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

      2. +-commutative 83.8%

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

      3. associate-*l/ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate--l+ 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

Alternative 2: 66.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+71}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-235}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= z -1.5e+71)
     (- x)
     (if (<= z -1.8e-50)
       t_0
       (if (<= z -4.5e-130)
         (/ x z)
         (if (<= z -4.8e-180)
           t_0
           (if (<= z 5e-235) (/ x z) (if (<= z 1.15e+39) t_0 (- x)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -1.5e+71) {
		tmp = -x;
	} else if (z <= -1.8e-50) {
		tmp = t_0;
	} else if (z <= -4.5e-130) {
		tmp = x / z;
	} else if (z <= -4.8e-180) {
		tmp = t_0;
	} else if (z <= 5e-235) {
		tmp = x / z;
	} else if (z <= 1.15e+39) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x / z)
    if (z <= (-1.5d+71)) then
        tmp = -x
    else if (z <= (-1.8d-50)) then
        tmp = t_0
    else if (z <= (-4.5d-130)) then
        tmp = x / z
    else if (z <= (-4.8d-180)) then
        tmp = t_0
    else if (z <= 5d-235) then
        tmp = x / z
    else if (z <= 1.15d+39) then
        tmp = t_0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -1.5e+71) {
		tmp = -x;
	} else if (z <= -1.8e-50) {
		tmp = t_0;
	} else if (z <= -4.5e-130) {
		tmp = x / z;
	} else if (z <= -4.8e-180) {
		tmp = t_0;
	} else if (z <= 5e-235) {
		tmp = x / z;
	} else if (z <= 1.15e+39) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if z <= -1.5e+71:
		tmp = -x
	elif z <= -1.8e-50:
		tmp = t_0
	elif z <= -4.5e-130:
		tmp = x / z
	elif z <= -4.8e-180:
		tmp = t_0
	elif z <= 5e-235:
		tmp = x / z
	elif z <= 1.15e+39:
		tmp = t_0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (z <= -1.5e+71)
		tmp = Float64(-x);
	elseif (z <= -1.8e-50)
		tmp = t_0;
	elseif (z <= -4.5e-130)
		tmp = Float64(x / z);
	elseif (z <= -4.8e-180)
		tmp = t_0;
	elseif (z <= 5e-235)
		tmp = Float64(x / z);
	elseif (z <= 1.15e+39)
		tmp = t_0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	tmp = 0.0;
	if (z <= -1.5e+71)
		tmp = -x;
	elseif (z <= -1.8e-50)
		tmp = t_0;
	elseif (z <= -4.5e-130)
		tmp = x / z;
	elseif (z <= -4.8e-180)
		tmp = t_0;
	elseif (z <= 5e-235)
		tmp = x / z;
	elseif (z <= 1.15e+39)
		tmp = t_0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.5e+71], (-x), If[LessEqual[z, -1.8e-50], t$95$0, If[LessEqual[z, -4.5e-130], N[(x / z), $MachinePrecision], If[LessEqual[z, -4.8e-180], t$95$0, If[LessEqual[z, 5e-235], N[(x / z), $MachinePrecision], If[LessEqual[z, 1.15e+39], t$95$0, (-x)]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.50000000000000006e71 or 1.15000000000000006e39 < z

   1. Initial program 67.9%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 88.2%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 88.2%

         \[\leadsto \color{blue}{-x} \]

   6. Simplified 88.2%

      \[\leadsto \color{blue}{-x} \]

   if -1.50000000000000006e71 < z < -1.7999999999999999e-50 or -4.5e-130 < z < -4.79999999999999959e-180 or 4.9999999999999998e-235 < z < 1.15000000000000006e39

   1. Initial program 98.7%

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

      1. associate-/l* 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in y around inf 64.1%

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

      1. associate-/l* 68.1%

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

   6. Simplified 68.1%

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

      1. clear-num 68.1%

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

      2. associate-/r/ 68.1%

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

      3. clear-num 68.1%

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

   8. Applied egg-rr 68.1%

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

   if -1.7999999999999999e-50 < z < -4.5e-130 or -4.79999999999999959e-180 < z < 4.9999999999999998e-235

   1. Initial program 99.9%

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

      1. associate-/l* 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in z around 0 98.2%

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

   5. Taylor expanded in y around 0 77.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+71}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-50}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-235}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+71}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \frac{y + 1}{z}\\ \mathbf{elif}\;z \leq -12000000:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+38}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.5e+71)
   (- x)
   (if (<= z -3.2e+20)
     (* x (/ (+ y 1.0) z))
     (if (<= z -12000000.0)
       (- (/ x z) x)
       (if (<= z 8e+38) (/ (+ x (* x y)) z) (- x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.5e+71) {
		tmp = -x;
	} else if (z <= -3.2e+20) {
		tmp = x * ((y + 1.0) / z);
	} else if (z <= -12000000.0) {
		tmp = (x / z) - x;
	} else if (z <= 8e+38) {
		tmp = (x + (x * y)) / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.5d+71)) then
        tmp = -x
    else if (z <= (-3.2d+20)) then
        tmp = x * ((y + 1.0d0) / z)
    else if (z <= (-12000000.0d0)) then
        tmp = (x / z) - x
    else if (z <= 8d+38) then
        tmp = (x + (x * y)) / z
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.5e+71) {
		tmp = -x;
	} else if (z <= -3.2e+20) {
		tmp = x * ((y + 1.0) / z);
	} else if (z <= -12000000.0) {
		tmp = (x / z) - x;
	} else if (z <= 8e+38) {
		tmp = (x + (x * y)) / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.5e+71:
		tmp = -x
	elif z <= -3.2e+20:
		tmp = x * ((y + 1.0) / z)
	elif z <= -12000000.0:
		tmp = (x / z) - x
	elif z <= 8e+38:
		tmp = (x + (x * y)) / z
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.5e+71)
		tmp = Float64(-x);
	elseif (z <= -3.2e+20)
		tmp = Float64(x * Float64(Float64(y + 1.0) / z));
	elseif (z <= -12000000.0)
		tmp = Float64(Float64(x / z) - x);
	elseif (z <= 8e+38)
		tmp = Float64(Float64(x + Float64(x * y)) / z);
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.5e+71)
		tmp = -x;
	elseif (z <= -3.2e+20)
		tmp = x * ((y + 1.0) / z);
	elseif (z <= -12000000.0)
		tmp = (x / z) - x;
	elseif (z <= 8e+38)
		tmp = (x + (x * y)) / z;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.5e+71], (-x), If[LessEqual[z, -3.2e+20], N[(x * N[(N[(y + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -12000000.0], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[z, 8e+38], N[(N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], (-x)]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.50000000000000006e71 or 7.99999999999999982e38 < z

   1. Initial program 67.9%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 88.2%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 88.2%

         \[\leadsto \color{blue}{-x} \]

   6. Simplified 88.2%

      \[\leadsto \color{blue}{-x} \]

   if -1.50000000000000006e71 < z < -3.2e20

   1. Initial program 90.9%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in z around 0 82.9%

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

      1. clear-num 83.0%

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

      2. associate-/r/ 83.0%

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

      3. clear-num 83.3%

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

      4. +-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

   if -3.2e20 < z < -1.2e7

   1. Initial program 99.6%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.6%

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

      3. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-rgt1-in 99.6%

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

      4. cancel-sign-sub-inv 99.6%

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

      5. div-sub 100.0%

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

      6. associate-*l/ 100.0%

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

      7. *-inverses 100.0%

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

      8. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -1.2e7 < z < 7.99999999999999982e38

   1. Initial program 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. fma-def 99.8%

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

      3. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 96.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+71}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \frac{y + 1}{z}\\ \mathbf{elif}\;z \leq -12000000:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+38}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+72}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{\frac{z}{y + 1}}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.2e+72) (- x) (if (<= z 1.4e+39) (/ x (/ z (+ y 1.0))) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e+72) {
		tmp = -x;
	} else if (z <= 1.4e+39) {
		tmp = x / (z / (y + 1.0));
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.2d+72)) then
        tmp = -x
    else if (z <= 1.4d+39) then
        tmp = x / (z / (y + 1.0d0))
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e+72) {
		tmp = -x;
	} else if (z <= 1.4e+39) {
		tmp = x / (z / (y + 1.0));
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.2e+72:
		tmp = -x
	elif z <= 1.4e+39:
		tmp = x / (z / (y + 1.0))
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.2e+72)
		tmp = Float64(-x);
	elseif (z <= 1.4e+39)
		tmp = Float64(x / Float64(z / Float64(y + 1.0)));
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.2e+72)
		tmp = -x;
	elseif (z <= 1.4e+39)
		tmp = x / (z / (y + 1.0));
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.2e+72], (-x), If[LessEqual[z, 1.4e+39], N[(x / N[(z / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e72 or 1.40000000000000001e39 < z

   1. Initial program 67.9%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 88.2%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 88.2%

         \[\leadsto \color{blue}{-x} \]

   6. Simplified 88.2%

      \[\leadsto \color{blue}{-x} \]

   if -2.2e72 < z < 1.40000000000000001e39

   1. Initial program 99.2%

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

      1. associate-/l* 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in z around 0 89.8%

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{1 + y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+72}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{\frac{z}{y + 1}}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 5: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+129}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.32e+129) (- x) (* (/ x z) (- (+ y 1.0) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.32e+129) {
		tmp = -x;
	} else {
		tmp = (x / z) * ((y + 1.0) - 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 (z <= (-1.32d+129)) then
        tmp = -x
    else
        tmp = (x / z) * ((y + 1.0d0) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.32e+129) {
		tmp = -x;
	} else {
		tmp = (x / z) * ((y + 1.0) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.32e+129:
		tmp = -x
	else:
		tmp = (x / z) * ((y + 1.0) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.32e+129)
		tmp = Float64(-x);
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y + 1.0) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.32e+129)
		tmp = -x;
	else
		tmp = (x / z) * ((y + 1.0) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.32e+129], (-x), N[(N[(x / z), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e129

   1. Initial program 64.3%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 93.7%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 93.7%

         \[\leadsto \color{blue}{-x} \]

   6. Simplified 93.7%

      \[\leadsto \color{blue}{-x} \]

   if -1.32e129 < z

   1. Initial program 92.2%

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

      1. associate-/l* 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in x around 0 92.2%

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

      1. associate--l+ 92.2%

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

      2. +-commutative 92.2%

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

      3. associate-*l/ 95.4%

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

      4. +-commutative 95.4%

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

      5. associate--l+ 95.4%

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

   6. Simplified 95.4%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+129}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

Alternative 6: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -85000000 \lor \neg \left(y \leq 6.5 \cdot 10^{+79}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -85000000.0) (not (<= y 6.5e+79))) (* y (/ x z)) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -85000000.0) || !(y <= 6.5e+79)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	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 <= (-85000000.0d0)) .or. (.not. (y <= 6.5d+79))) then
        tmp = y * (x / z)
    else
        tmp = (x / z) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -85000000.0) || !(y <= 6.5e+79)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -85000000.0) or not (y <= 6.5e+79):
		tmp = y * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -85000000.0) || !(y <= 6.5e+79))
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -85000000.0) || ~((y <= 6.5e+79)))
		tmp = y * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -85000000.0], N[Not[LessEqual[y, 6.5e+79]], $MachinePrecision]], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5e7 or 6.49999999999999954e79 < y

   1. Initial program 89.8%

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

      1. associate-/l* 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in y around inf 77.3%

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

      1. associate-/l* 78.7%

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

   6. Simplified 78.7%

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

      1. clear-num 78.7%

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

      2. associate-/r/ 78.7%

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

      3. clear-num 78.7%

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

   8. Applied egg-rr 78.7%

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

   if -8.5e7 < y < 6.49999999999999954e79

   1. Initial program 85.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.8%

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

      3. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 80.3%

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

      1. sub-neg 80.3%

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

      2. +-commutative 80.3%

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

      3. distribute-rgt1-in 80.3%

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

      4. cancel-sign-sub-inv 80.3%

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

      5. div-sub 80.3%

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

      6. associate-*l/ 94.3%

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

      7. *-inverses 94.3%

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

      8. *-lft-identity 94.3%

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

   8. Simplified 94.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -85000000 \lor \neg \left(y \leq 6.5 \cdot 10^{+79}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 7: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -17500000:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -17500000.0)
   (* y (/ x z))
   (if (<= y 1.35e+80) (- (/ x z) x) (/ y (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -17500000.0) {
		tmp = y * (x / z);
	} else if (y <= 1.35e+80) {
		tmp = (x / z) - x;
	} else {
		tmp = y / (z / x);
	}
	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 <= (-17500000.0d0)) then
        tmp = y * (x / z)
    else if (y <= 1.35d+80) then
        tmp = (x / z) - x
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -17500000.0) {
		tmp = y * (x / z);
	} else if (y <= 1.35e+80) {
		tmp = (x / z) - x;
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -17500000.0:
		tmp = y * (x / z)
	elif y <= 1.35e+80:
		tmp = (x / z) - x
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -17500000.0)
		tmp = Float64(y * Float64(x / z));
	elseif (y <= 1.35e+80)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -17500000.0)
		tmp = y * (x / z);
	elseif (y <= 1.35e+80)
		tmp = (x / z) - x;
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -17500000.0], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+80], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.75e7

   1. Initial program 94.1%

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

      1. associate-/l* 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in y around inf 79.1%

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

      1. associate-/l* 81.0%

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

   6. Simplified 81.0%

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

      1. clear-num 81.0%

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

      2. associate-/r/ 81.0%

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

      3. clear-num 81.1%

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

   8. Applied egg-rr 81.1%

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

   if -1.75e7 < y < 1.34999999999999991e80

   1. Initial program 85.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.8%

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

      3. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 80.3%

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

      1. sub-neg 80.3%

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

      2. +-commutative 80.3%

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

      3. distribute-rgt1-in 80.3%

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

      4. cancel-sign-sub-inv 80.3%

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

      5. div-sub 80.3%

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

      6. associate-*l/ 94.3%

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

      7. *-inverses 94.3%

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

      8. *-lft-identity 94.3%

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

   8. Simplified 94.3%

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

   if 1.34999999999999991e80 < y

   1. Initial program 85.1%

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

      1. associate-/l* 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in y around inf 75.3%

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

      1. associate-/l* 76.1%

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

   6. Simplified 76.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -17500000:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 8: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x * (((y - z) + 1.0) / 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) + 1.0d0) / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. associate-/l* 97.7%

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

3. Simplified 97.7%

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

   1. clear-num 97.4%

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

   2. associate-/r/ 96.8%

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

   3. clear-num 96.9%

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

5. Applied egg-rr 96.9%

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

6. Final simplification 96.9%

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

Alternative 9: 65.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -960:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.25:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -960.0) (- x) (if (<= z 1.25) (/ x z) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -960.0) {
		tmp = -x;
	} else if (z <= 1.25) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-960.0d0)) then
        tmp = -x
    else if (z <= 1.25d0) then
        tmp = x / z
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -960.0) {
		tmp = -x;
	} else if (z <= 1.25) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -960.0:
		tmp = -x
	elif z <= 1.25:
		tmp = x / z
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -960.0)
		tmp = Float64(-x);
	elseif (z <= 1.25)
		tmp = Float64(x / z);
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -960.0)
		tmp = -x;
	elseif (z <= 1.25)
		tmp = x / z;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -960.0], (-x), If[LessEqual[z, 1.25], N[(x / z), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if z < -960 or 1.25 < z

   1. Initial program 73.4%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 77.4%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 77.4%

         \[\leadsto \color{blue}{-x} \]

   6. Simplified 77.4%

      \[\leadsto \color{blue}{-x} \]

   if -960 < z < 1.25

   1. Initial program 99.8%

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

      1. associate-/l* 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in z around 0 94.9%

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

   5. Taylor expanded in y around 0 57.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -960:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.25:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 10: 39.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return -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 = -x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-x)

Derivation

1. Initial program 87.3%

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

   1. associate-/l* 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in z around inf 38.2%

   \[\leadsto \color{blue}{-1 \cdot x} \]
5. Step-by-step derivation

   1. neg-mul-1 38.2%

      \[\leadsto \color{blue}{-x} \]

6. Simplified 38.2%

   \[\leadsto \color{blue}{-x} \]

7. Final simplification 38.2%

   \[\leadsto -x \]

Developer target: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ 1.0 y) (/ x z)) x)))
   (if (< x -2.71483106713436e-162)
     t_0
     (if (< x 3.874108816439546e-197)
       (* (* x (+ (- y z) 1.0)) (/ 1.0 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / 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 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / 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 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 + y) * (x / z)) - x
	tmp = 0
	if x < -2.71483106713436e-162:
		tmp = t_0
	elif x < 3.874108816439546e-197:
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 + y) * Float64(x / z)) - x)
	tmp = 0.0
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) * Float64(1.0 / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 + y) * (x / z)) - x;
	tmp = 0.0;
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[Less[x, -2.71483106713436e-162], t$95$0, If[Less[x, 3.874108816439546e-197], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2023223 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))