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

Percentage Accurate: 88.0% → 99.8%

Time: 8.2s

Alternatives: 12

Speedup: 0.6×

• 20.7% 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: 88.0% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.42 \cdot 10^{-5} \lor \neg \left(z \leq 1.85 \cdot 10^{-30}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.42e-5) (not (<= z 1.85e-30)))
   (* x (+ -1.0 (/ (+ 1.0 y) z)))
   (/ (+ x (* x y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.42e-5) || !(z <= 1.85e-30)) {
		tmp = x * (-1.0 + ((1.0 + y) / z));
	} else {
		tmp = (x + (x * y)) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.42d-5)) .or. (.not. (z <= 1.85d-30))) then
        tmp = x * ((-1.0d0) + ((1.0d0 + y) / z))
    else
        tmp = (x + (x * y)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.42e-5) || !(z <= 1.85e-30)) {
		tmp = x * (-1.0 + ((1.0 + y) / z));
	} else {
		tmp = (x + (x * y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.42e-5) or not (z <= 1.85e-30):
		tmp = x * (-1.0 + ((1.0 + y) / z))
	else:
		tmp = (x + (x * y)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.42e-5) || !(z <= 1.85e-30))
		tmp = Float64(x * Float64(-1.0 + Float64(Float64(1.0 + y) / z)));
	else
		tmp = Float64(Float64(x + Float64(x * y)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.42e-5) || ~((z <= 1.85e-30)))
		tmp = x * (-1.0 + ((1.0 + y) / z));
	else
		tmp = (x + (x * y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.42e-5], N[Not[LessEqual[z, 1.85e-30]], $MachinePrecision]], N[(x * N[(-1.0 + N[(N[(1.0 + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.42e-5 or 1.8500000000000002e-30 < z

   1. Initial program 67.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 67.9%

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

      1. associate--l+ 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-*r/ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate--l+ 99.9%

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

      6. div-sub 99.9%

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

      7. sub-neg 99.9%

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

      8. *-inverses 99.9%

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

      9. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   if -1.42e-5 < z < 1.8500000000000002e-30

   1. Initial program 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. fma-def 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.42 \cdot 10^{-5} \lor \neg \left(z \leq 1.85 \cdot 10^{-30}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+47}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+16} \lor \neg \left(y \leq 1.65 \cdot 10^{+15}\right) \land \left(y \leq 1.86 \cdot 10^{+56} \lor \neg \left(y \leq 4.6 \cdot 10^{+116}\right)\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= y -1.15e+69)
     t_0
     (if (<= y -5.8e+47)
       (- x)
       (if (or (<= y -6.5e+16)
               (and (not (<= y 1.65e+15))
                    (or (<= y 1.86e+56) (not (<= y 4.6e+116)))))
         t_0
         (- (/ x z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (y <= -1.15e+69) {
		tmp = t_0;
	} else if (y <= -5.8e+47) {
		tmp = -x;
	} else if ((y <= -6.5e+16) || (!(y <= 1.65e+15) && ((y <= 1.86e+56) || !(y <= 4.6e+116)))) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y * (x / z)
    if (y <= (-1.15d+69)) then
        tmp = t_0
    else if (y <= (-5.8d+47)) then
        tmp = -x
    else if ((y <= (-6.5d+16)) .or. (.not. (y <= 1.65d+15)) .and. (y <= 1.86d+56) .or. (.not. (y <= 4.6d+116))) then
        tmp = t_0
    else
        tmp = (x / z) - 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 (y <= -1.15e+69) {
		tmp = t_0;
	} else if (y <= -5.8e+47) {
		tmp = -x;
	} else if ((y <= -6.5e+16) || (!(y <= 1.65e+15) && ((y <= 1.86e+56) || !(y <= 4.6e+116)))) {
		tmp = t_0;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if y <= -1.15e+69:
		tmp = t_0
	elif y <= -5.8e+47:
		tmp = -x
	elif (y <= -6.5e+16) or (not (y <= 1.65e+15) and ((y <= 1.86e+56) or not (y <= 4.6e+116))):
		tmp = t_0
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (y <= -1.15e+69)
		tmp = t_0;
	elseif (y <= -5.8e+47)
		tmp = Float64(-x);
	elseif ((y <= -6.5e+16) || (!(y <= 1.65e+15) && ((y <= 1.86e+56) || !(y <= 4.6e+116))))
		tmp = t_0;
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	tmp = 0.0;
	if (y <= -1.15e+69)
		tmp = t_0;
	elseif (y <= -5.8e+47)
		tmp = -x;
	elseif ((y <= -6.5e+16) || (~((y <= 1.65e+15)) && ((y <= 1.86e+56) || ~((y <= 4.6e+116)))))
		tmp = t_0;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e+69], t$95$0, If[LessEqual[y, -5.8e+47], (-x), If[Or[LessEqual[y, -6.5e+16], And[N[Not[LessEqual[y, 1.65e+15]], $MachinePrecision], Or[LessEqual[y, 1.86e+56], N[Not[LessEqual[y, 4.6e+116]], $MachinePrecision]]]], t$95$0, N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15000000000000008e69 or -5.79999999999999961e47 < y < -6.5e16 or 1.65e15 < y < 1.86000000000000007e56 or 4.5999999999999999e116 < y

   1. Initial program 90.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 79.6%

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

      1. associate-/l* 73.4%

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

      2. associate-/r/ 83.1%

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

   5. Simplified 83.1%

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

   if -1.15000000000000008e69 < y < -5.79999999999999961e47

   1. Initial program 42.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   if -6.5e16 < y < 1.65e15 or 1.86000000000000007e56 < y < 4.5999999999999999e116

   1. Initial program 81.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 77.4%

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

   4. Taylor expanded in z around 0 96.2%

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

      1. neg-mul-1 96.2%

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

      2. +-commutative 96.2%

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

      3. unsub-neg 96.2%

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

   6. Simplified 96.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+47}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+16} \lor \neg \left(y \leq 1.65 \cdot 10^{+15}\right) \land \left(y \leq 1.86 \cdot 10^{+56} \lor \neg \left(y \leq 4.6 \cdot 10^{+116}\right)\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{+50}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+16} \lor \neg \left(y \leq 1.25 \cdot 10^{+55}\right) \land y \leq 9.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= y -1.1e+69)
     t_0
     (if (<= y -1.3e+50)
       (- x)
       (if (<= y -1.45e+16)
         (/ y (/ z x))
         (if (or (<= y 1.7e+16) (and (not (<= y 1.25e+55)) (<= y 9.2e+115)))
           (- (/ x z) x)
           t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (y <= -1.1e+69) {
		tmp = t_0;
	} else if (y <= -1.3e+50) {
		tmp = -x;
	} else if (y <= -1.45e+16) {
		tmp = y / (z / x);
	} else if ((y <= 1.7e+16) || (!(y <= 1.25e+55) && (y <= 9.2e+115))) {
		tmp = (x / z) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x / z)
    if (y <= (-1.1d+69)) then
        tmp = t_0
    else if (y <= (-1.3d+50)) then
        tmp = -x
    else if (y <= (-1.45d+16)) then
        tmp = y / (z / x)
    else if ((y <= 1.7d+16) .or. (.not. (y <= 1.25d+55)) .and. (y <= 9.2d+115)) then
        tmp = (x / z) - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (y <= -1.1e+69) {
		tmp = t_0;
	} else if (y <= -1.3e+50) {
		tmp = -x;
	} else if (y <= -1.45e+16) {
		tmp = y / (z / x);
	} else if ((y <= 1.7e+16) || (!(y <= 1.25e+55) && (y <= 9.2e+115))) {
		tmp = (x / z) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if y <= -1.1e+69:
		tmp = t_0
	elif y <= -1.3e+50:
		tmp = -x
	elif y <= -1.45e+16:
		tmp = y / (z / x)
	elif (y <= 1.7e+16) or (not (y <= 1.25e+55) and (y <= 9.2e+115)):
		tmp = (x / z) - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (y <= -1.1e+69)
		tmp = t_0;
	elseif (y <= -1.3e+50)
		tmp = Float64(-x);
	elseif (y <= -1.45e+16)
		tmp = Float64(y / Float64(z / x));
	elseif ((y <= 1.7e+16) || (!(y <= 1.25e+55) && (y <= 9.2e+115)))
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	tmp = 0.0;
	if (y <= -1.1e+69)
		tmp = t_0;
	elseif (y <= -1.3e+50)
		tmp = -x;
	elseif (y <= -1.45e+16)
		tmp = y / (z / x);
	elseif ((y <= 1.7e+16) || (~((y <= 1.25e+55)) && (y <= 9.2e+115)))
		tmp = (x / z) - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.1e+69], t$95$0, If[LessEqual[y, -1.3e+50], (-x), If[LessEqual[y, -1.45e+16], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.7e+16], And[N[Not[LessEqual[y, 1.25e+55]], $MachinePrecision], LessEqual[y, 9.2e+115]]], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.1000000000000001e69 or 1.7e16 < y < 1.25000000000000011e55 or 9.20000000000000014e115 < y

   1. Initial program 89.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 78.5%

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

      1. associate-/l* 72.6%

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

      2. associate-/r/ 82.3%

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

   5. Simplified 82.3%

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

   if -1.1000000000000001e69 < y < -1.3000000000000001e50

   1. Initial program 42.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   if -1.3000000000000001e50 < y < -1.45e16

   1. Initial program 99.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 90.2%

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

      1. associate-/l* 80.9%

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

      2. associate-/r/ 90.1%

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

   5. Simplified 90.1%

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

      1. *-commutative 90.1%

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

      2. clear-num 89.9%

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

      3. un-div-inv 90.4%

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

   7. Applied egg-rr 90.4%

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

   if -1.45e16 < y < 1.7e16 or 1.25000000000000011e55 < y < 9.20000000000000014e115

   1. Initial program 81.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 77.4%

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

   4. Taylor expanded in z around 0 96.2%

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

      1. neg-mul-1 96.2%

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

      2. +-commutative 96.2%

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

      3. unsub-neg 96.2%

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

   6. Simplified 96.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{+50}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+16} \lor \neg \left(y \leq 1.25 \cdot 10^{+55}\right) \land y \leq 9.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -5.9 \cdot 10^{+23}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.18 \cdot 10^{-225}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= z -5.9e+23)
     (- x)
     (if (<= z -1.18e-225)
       t_0
       (if (<= z 1.85e-287)
         (/ x z)
         (if (<= z 6.4e-150) t_0 (if (<= z 3.3e+19) (/ x z) (- x))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -5.9e+23) {
		tmp = -x;
	} else if (z <= -1.18e-225) {
		tmp = t_0;
	} else if (z <= 1.85e-287) {
		tmp = x / z;
	} else if (z <= 6.4e-150) {
		tmp = t_0;
	} else if (z <= 3.3e+19) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y * (x / z)
    if (z <= (-5.9d+23)) then
        tmp = -x
    else if (z <= (-1.18d-225)) then
        tmp = t_0
    else if (z <= 1.85d-287) then
        tmp = x / z
    else if (z <= 6.4d-150) then
        tmp = t_0
    else if (z <= 3.3d+19) 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 t_0 = y * (x / z);
	double tmp;
	if (z <= -5.9e+23) {
		tmp = -x;
	} else if (z <= -1.18e-225) {
		tmp = t_0;
	} else if (z <= 1.85e-287) {
		tmp = x / z;
	} else if (z <= 6.4e-150) {
		tmp = t_0;
	} else if (z <= 3.3e+19) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if z <= -5.9e+23:
		tmp = -x
	elif z <= -1.18e-225:
		tmp = t_0
	elif z <= 1.85e-287:
		tmp = x / z
	elif z <= 6.4e-150:
		tmp = t_0
	elif z <= 3.3e+19:
		tmp = x / z
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (z <= -5.9e+23)
		tmp = Float64(-x);
	elseif (z <= -1.18e-225)
		tmp = t_0;
	elseif (z <= 1.85e-287)
		tmp = Float64(x / z);
	elseif (z <= 6.4e-150)
		tmp = t_0;
	elseif (z <= 3.3e+19)
		tmp = Float64(x / z);
	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 <= -5.9e+23)
		tmp = -x;
	elseif (z <= -1.18e-225)
		tmp = t_0;
	elseif (z <= 1.85e-287)
		tmp = x / z;
	elseif (z <= 6.4e-150)
		tmp = t_0;
	elseif (z <= 3.3e+19)
		tmp = x / z;
	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, -5.9e+23], (-x), If[LessEqual[z, -1.18e-225], t$95$0, If[LessEqual[z, 1.85e-287], N[(x / z), $MachinePrecision], If[LessEqual[z, 6.4e-150], t$95$0, If[LessEqual[z, 3.3e+19], N[(x / z), $MachinePrecision], (-x)]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.89999999999999987e23 or 3.3e19 < z

   1. Initial program 64.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 82.5%

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

      1. mul-1-neg 82.5%

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

   5. Simplified 82.5%

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

   if -5.89999999999999987e23 < z < -1.18e-225 or 1.85000000000000013e-287 < z < 6.3999999999999996e-150

   1. Initial program 98.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 70.8%

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

      1. associate-/l* 59.5%

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

      2. associate-/r/ 74.3%

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

   5. Simplified 74.3%

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

   if -1.18e-225 < z < 1.85000000000000013e-287 or 6.3999999999999996e-150 < z < 3.3e19

   1. Initial program 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. fma-def 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.9%

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

   6. Taylor expanded in y around 0 69.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{+23}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.18 \cdot 10^{-225}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-150}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+187}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq -1 \lor \neg \left(y \leq 0.0073\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e+187)
   (/ (* x y) z)
   (if (or (<= y -1.0) (not (<= y 0.0073)))
     (* x (+ -1.0 (/ y z)))
     (- (/ x z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e+187) {
		tmp = (x * y) / z;
	} else if ((y <= -1.0) || !(y <= 0.0073)) {
		tmp = x * (-1.0 + (y / 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 <= (-4.8d+187)) then
        tmp = (x * y) / z
    else if ((y <= (-1.0d0)) .or. (.not. (y <= 0.0073d0))) then
        tmp = x * ((-1.0d0) + (y / 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 <= -4.8e+187) {
		tmp = (x * y) / z;
	} else if ((y <= -1.0) || !(y <= 0.0073)) {
		tmp = x * (-1.0 + (y / z));
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.8e+187:
		tmp = (x * y) / z
	elif (y <= -1.0) or not (y <= 0.0073):
		tmp = x * (-1.0 + (y / z))
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.8e+187)
		tmp = Float64(Float64(x * y) / z);
	elseif ((y <= -1.0) || !(y <= 0.0073))
		tmp = Float64(x * Float64(-1.0 + Float64(y / 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 <= -4.8e+187)
		tmp = (x * y) / z;
	elseif ((y <= -1.0) || ~((y <= 0.0073)))
		tmp = x * (-1.0 + (y / z));
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.8e+187], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.0073]], $MachinePrecision]], N[(x * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.79999999999999971e187

   1. Initial program 95.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 92.1%

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

   if -4.79999999999999971e187 < y < -1 or 0.00730000000000000007 < y

   1. Initial program 85.0%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 91.0%

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

   4. Taylor expanded in y around inf 90.4%

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

      1. associate-*r/ 91.3%

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

   6. Simplified 91.3%

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

   7. Taylor expanded in x around 0 91.3%

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

   if -1 < y < 0.00730000000000000007

   1. Initial program 81.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 79.8%

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

   4. Taylor expanded in z around 0 98.6%

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

      1. neg-mul-1 98.6%

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

      2. +-commutative 98.6%

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

      3. unsub-neg 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+187}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq -1 \lor \neg \left(y \leq 0.0073\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+187}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 0.0073:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e+187)
   (/ (* x y) z)
   (if (<= y -1.0)
     (* x (+ -1.0 (/ y z)))
     (if (<= y 0.0073) (- (/ x z) x) (- (* x (/ y z)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+187) {
		tmp = (x * y) / z;
	} else if (y <= -1.0) {
		tmp = x * (-1.0 + (y / z));
	} else if (y <= 0.0073) {
		tmp = (x / z) - x;
	} else {
		tmp = (x * (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 <= (-4d+187)) then
        tmp = (x * y) / z
    else if (y <= (-1.0d0)) then
        tmp = x * ((-1.0d0) + (y / z))
    else if (y <= 0.0073d0) then
        tmp = (x / z) - x
    else
        tmp = (x * (y / z)) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+187) {
		tmp = (x * y) / z;
	} else if (y <= -1.0) {
		tmp = x * (-1.0 + (y / z));
	} else if (y <= 0.0073) {
		tmp = (x / z) - x;
	} else {
		tmp = (x * (y / z)) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e+187:
		tmp = (x * y) / z
	elif y <= -1.0:
		tmp = x * (-1.0 + (y / z))
	elif y <= 0.0073:
		tmp = (x / z) - x
	else:
		tmp = (x * (y / z)) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e+187)
		tmp = Float64(Float64(x * y) / z);
	elseif (y <= -1.0)
		tmp = Float64(x * Float64(-1.0 + Float64(y / z)));
	elseif (y <= 0.0073)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(Float64(x * Float64(y / z)) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4e+187)
		tmp = (x * y) / z;
	elseif (y <= -1.0)
		tmp = x * (-1.0 + (y / z));
	elseif (y <= 0.0073)
		tmp = (x / z) - x;
	else
		tmp = (x * (y / z)) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e+187], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, -1.0], N[(x * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0073], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.99999999999999963e187

   1. Initial program 95.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 92.1%

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

   if -3.99999999999999963e187 < y < -1

   1. Initial program 84.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 94.8%

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

   4. Taylor expanded in y around inf 94.8%

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

      1. associate-*r/ 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in x around 0 89.9%

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

   if -1 < y < 0.00730000000000000007

   1. Initial program 81.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 79.8%

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

   4. Taylor expanded in z around 0 98.6%

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

      1. neg-mul-1 98.6%

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

      2. +-commutative 98.6%

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

      3. unsub-neg 98.6%

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

   6. Simplified 98.6%

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

   if 0.00730000000000000007 < y

   1. Initial program 85.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 88.9%

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

   4. Taylor expanded in y around inf 88.0%

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

      1. associate-*r/ 92.0%

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

   6. Simplified 92.0%

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

      1. +-commutative 92.0%

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

      2. mul-1-neg 92.0%

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

      3. unsub-neg 92.0%

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

   8. Applied egg-rr 92.0%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+187}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 0.0073:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.1)
   (- (* x (/ y z)) x)
   (if (<= z 1.0) (/ (+ x (* x y)) z) (* x (+ -1.0 (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.1) {
		tmp = (x * (y / z)) - x;
	} else if (z <= 1.0) {
		tmp = (x + (x * y)) / z;
	} else {
		tmp = x * (-1.0 + (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 (z <= (-1.1d0)) then
        tmp = (x * (y / z)) - x
    else if (z <= 1.0d0) then
        tmp = (x + (x * y)) / z
    else
        tmp = x * ((-1.0d0) + (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.1) {
		tmp = (x * (y / z)) - x;
	} else if (z <= 1.0) {
		tmp = (x + (x * y)) / z;
	} else {
		tmp = x * (-1.0 + (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.1:
		tmp = (x * (y / z)) - x
	elif z <= 1.0:
		tmp = (x + (x * y)) / z
	else:
		tmp = x * (-1.0 + (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.1)
		tmp = Float64(Float64(x * Float64(y / z)) - x);
	elseif (z <= 1.0)
		tmp = Float64(Float64(x + Float64(x * y)) / z);
	else
		tmp = Float64(x * Float64(-1.0 + Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.1)
		tmp = (x * (y / z)) - x;
	elseif (z <= 1.0)
		tmp = (x + (x * y)) / z;
	else
		tmp = x * (-1.0 + (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.1], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1000000000000001

   1. Initial program 66.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 94.3%

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

   4. Taylor expanded in y around inf 93.0%

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

      1. associate-*r/ 98.6%

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

   6. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

   8. Applied egg-rr 98.6%

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

   if -1.1000000000000001 < z < 1

   1. Initial program 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. fma-def 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.9%

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

   if 1 < z

   1. Initial program 66.3%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 87.4%

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

   4. Taylor expanded in y around inf 87.4%

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

      1. associate-*r/ 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \left(1 - z\right)}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.2e-8) {
		tmp = ((x * (1.0 + y)) / z) - x;
	} else {
		tmp = (y + (1.0 - z)) / (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 (x <= 3.2d-8) then
        tmp = ((x * (1.0d0 + y)) / z) - x
    else
        tmp = (y + (1.0d0 - z)) / (z / x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002e-8

   1. Initial program 91.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 96.9%

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

   if 3.2000000000000002e-8 < x

   1. Initial program 63.8%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-+l+ 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \left(1 - z\right)}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.2e-62)
   (/ (* x (+ 1.0 (- y z))) z)
   (* x (+ -1.0 (/ (+ 1.0 y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.2e-62) {
		tmp = (x * (1.0 + (y - z))) / z;
	} else {
		tmp = x * (-1.0 + ((1.0 + 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 (x <= 2.2d-62) then
        tmp = (x * (1.0d0 + (y - z))) / z
    else
        tmp = x * ((-1.0d0) + ((1.0d0 + y) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.2e-62) {
		tmp = (x * (1.0 + (y - z))) / z;
	} else {
		tmp = x * (-1.0 + ((1.0 + y) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.2e-62:
		tmp = (x * (1.0 + (y - z))) / z
	else:
		tmp = x * (-1.0 + ((1.0 + y) / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.2e-62)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(y - z))) / z);
	else
		tmp = Float64(x * Float64(-1.0 + Float64(Float64(1.0 + y) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.2e-62)
		tmp = (x * (1.0 + (y - z))) / z;
	else
		tmp = x * (-1.0 + ((1.0 + y) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.2e-62], N[(N[(x * N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(-1.0 + N[(N[(1.0 + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000017e-62

   1. Initial program 90.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   if 2.20000000000000017e-62 < x

   1. Initial program 66.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 66.9%

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

      1. associate--l+ 66.9%

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

      2. +-commutative 66.9%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--l+ 99.8%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \left(1 - z\right)}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e-8) (/ (* x (+ 1.0 (- y z))) z) (/ (+ y (- 1.0 z)) (/ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e-8) {
		tmp = (x * (1.0 + (y - z))) / z;
	} else {
		tmp = (y + (1.0 - z)) / (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 (x <= 2d-8) then
        tmp = (x * (1.0d0 + (y - z))) / z
    else
        tmp = (y + (1.0d0 - z)) / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e-8) {
		tmp = (x * (1.0 + (y - z))) / z;
	} else {
		tmp = (y + (1.0 - z)) / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2e-8:
		tmp = (x * (1.0 + (y - z))) / z
	else:
		tmp = (y + (1.0 - z)) / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e-8)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(y - z))) / z);
	else
		tmp = Float64(Float64(y + Float64(1.0 - z)) / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2e-8)
		tmp = (x * (1.0 + (y - z))) / z;
	else
		tmp = (y + (1.0 - z)) / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e-8], N[(N[(x * N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y + N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-8

   1. Initial program 91.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   if 2e-8 < x

   1. Initial program 63.8%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-+l+ 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \left(1 - z\right)}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 3.3 \cdot 10^{+19}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 3.3e+19))) (- x) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 3.3e+19)) {
		tmp = -x;
	} else {
		tmp = 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 ((z <= (-1.0d0)) .or. (.not. (z <= 3.3d+19))) then
        tmp = -x
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 3.3e+19)) {
		tmp = -x;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 3.3e+19):
		tmp = -x
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 3.3e+19))
		tmp = Float64(-x);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 3.3e+19)))
		tmp = -x;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 3.3e+19]], $MachinePrecision]], (-x), N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 3.3e19 < z

   1. Initial program 66.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 78.1%

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

      1. mul-1-neg 78.1%

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

   5. Simplified 78.1%

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

   if -1 < z < 3.3e19

   1. Initial program 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. fma-def 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.9%

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

   6. Taylor expanded in y around 0 54.2%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 3.3 \cdot 10^{+19}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.0% 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 84.2%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf 38.1%

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

   1. mul-1-neg 38.1%

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

5. Simplified 38.1%

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

6. Final simplification 38.1%

   \[\leadsto -x \]
7. Add Preprocessing

Developer target: 99.4% accurate, 0.4× 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 2024029 
(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))