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

Percentage Accurate: 87.7% → 99.7%

Time: 5.2s

Alternatives: 10

Speedup: 0.8×

• 21.1% 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: 87.7% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+36} \lor \neg \left(z \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5e+36) (not (<= z 2e+20)))
   (* x (+ (/ y z) -1.0))
   (/ (fma x (- y z) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5e+36) || !(z <= 2e+20)) {
		tmp = x * ((y / z) + -1.0);
	} else {
		tmp = fma(x, (y - z), x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5e+36) || !(z <= 2e+20))
		tmp = Float64(x * Float64(Float64(y / z) + -1.0));
	else
		tmp = Float64(fma(x, Float64(y - z), x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5e+36], N[Not[LessEqual[z, 2e+20]], $MachinePrecision]], N[(x * N[(N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y - z), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.99999999999999977e36 or 2e20 < z

   1. Initial program 68.6%

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

   2. Taylor expanded in x around 0 68.6%

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

      1. associate--l+ 68.6%

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

      2. +-commutative 68.6%

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

      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%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around inf 99.9%

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

   if -4.99999999999999977e36 < z < 2e20

   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%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+36} \lor \neg \left(z \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-16} \lor \neg \left(z \leq 2.45 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2e-16) (not (<= z 2.45e-51)))
   (* x (+ -1.0 (/ (+ y 1.0) z)))
   (/ (+ x (* x y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2e-16) || !(z <= 2.45e-51)) {
		tmp = x * (-1.0 + ((y + 1.0) / 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 <= (-5.2d-16)) .or. (.not. (z <= 2.45d-51))) then
        tmp = x * ((-1.0d0) + ((y + 1.0d0) / 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 <= -5.2e-16) || !(z <= 2.45e-51)) {
		tmp = x * (-1.0 + ((y + 1.0) / z));
	} else {
		tmp = (x + (x * y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.2e-16) or not (z <= 2.45e-51):
		tmp = x * (-1.0 + ((y + 1.0) / z))
	else:
		tmp = (x + (x * y)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2e-16) || !(z <= 2.45e-51))
		tmp = Float64(x * Float64(-1.0 + Float64(Float64(y + 1.0) / 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 <= -5.2e-16) || ~((z <= 2.45e-51)))
		tmp = x * (-1.0 + ((y + 1.0) / z));
	else
		tmp = (x + (x * y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2e-16], N[Not[LessEqual[z, 2.45e-51]], $MachinePrecision]], N[(x * N[(-1.0 + N[(N[(y + 1.0), $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 < -5.1999999999999997e-16 or 2.44999999999999987e-51 < z

   1. Initial program 74.6%

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

   2. Taylor expanded in x around 0 74.6%

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

      1. associate--l+ 74.6%

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

      2. +-commutative 74.6%

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

      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%

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

   4. Simplified 99.9%

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

   if -5.1999999999999997e-16 < z < 2.44999999999999987e-51

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. fma-def 100.0%

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

      3. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-16} \lor \neg \left(z \leq 2.45 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+36} \lor \neg \left(z \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.05e+36) (not (<= z 2e+20)))
   (* x (+ (/ y z) -1.0))
   (/ (* x (+ (- y z) 1.0)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05e+36) || !(z <= 2e+20)) {
		tmp = x * ((y / z) + -1.0);
	} else {
		tmp = (x * ((y - z) + 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 <= (-1.05d+36)) .or. (.not. (z <= 2d+20))) then
        tmp = x * ((y / z) + (-1.0d0))
    else
        tmp = (x * ((y - z) + 1.0d0)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05e+36) || !(z <= 2e+20)) {
		tmp = x * ((y / z) + -1.0);
	} else {
		tmp = (x * ((y - z) + 1.0)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.05e+36) or not (z <= 2e+20):
		tmp = x * ((y / z) + -1.0)
	else:
		tmp = (x * ((y - z) + 1.0)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.05e+36) || !(z <= 2e+20))
		tmp = Float64(x * Float64(Float64(y / z) + -1.0));
	else
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.05e+36) || ~((z <= 2e+20)))
		tmp = x * ((y / z) + -1.0);
	else
		tmp = (x * ((y - z) + 1.0)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.05e+36], N[Not[LessEqual[z, 2e+20]], $MachinePrecision]], N[(x * N[(N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05000000000000002e36 or 2e20 < z

   1. Initial program 68.6%

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

   2. Taylor expanded in x around 0 68.6%

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

      1. associate--l+ 68.6%

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

      2. +-commutative 68.6%

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

      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%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around inf 99.9%

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

   if -1.05000000000000002e36 < z < 2e20

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+36} \lor \neg \left(z \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \end{array} \]

Alternative 4: 64.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-291}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (- x)
   (if (<= z 4.7e-291)
     (/ x z)
     (if (<= z 2.9e-114) (* y (/ x z)) (if (<= z 1.0) (/ x z) (- x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -x;
	} else if (z <= 4.7e-291) {
		tmp = x / z;
	} else if (z <= 2.9e-114) {
		tmp = y * (x / z);
	} else if (z <= 1.0) {
		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 <= (-1.0d0)) then
        tmp = -x
    else if (z <= 4.7d-291) then
        tmp = x / z
    else if (z <= 2.9d-114) then
        tmp = y * (x / z)
    else if (z <= 1.0d0) 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 <= -1.0) {
		tmp = -x;
	} else if (z <= 4.7e-291) {
		tmp = x / z;
	} else if (z <= 2.9e-114) {
		tmp = y * (x / z);
	} else if (z <= 1.0) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = -x
	elif z <= 4.7e-291:
		tmp = x / z
	elif z <= 2.9e-114:
		tmp = y * (x / z)
	elif z <= 1.0:
		tmp = x / z
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(-x);
	elseif (z <= 4.7e-291)
		tmp = Float64(x / z);
	elseif (z <= 2.9e-114)
		tmp = Float64(y * Float64(x / z));
	elseif (z <= 1.0)
		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 <= -1.0)
		tmp = -x;
	elseif (z <= 4.7e-291)
		tmp = x / z;
	elseif (z <= 2.9e-114)
		tmp = y * (x / z);
	elseif (z <= 1.0)
		tmp = x / z;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], (-x), If[LessEqual[z, 4.7e-291], N[(x / z), $MachinePrecision], If[LessEqual[z, 2.9e-114], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x / z), $MachinePrecision], (-x)]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1 < z

   1. Initial program 70.8%

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

   2. Taylor expanded in z around inf 77.3%

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

      1. mul-1-neg 77.3%

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

   4. Simplified 77.3%

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

   if -1 < z < 4.6999999999999998e-291 or 2.89999999999999997e-114 < z < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 64.2%

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

      1. associate-/l* 64.2%

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

   4. Simplified 64.2%

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

   5. Taylor expanded in z around 0 61.6%

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

   if 4.6999999999999998e-291 < z < 2.89999999999999997e-114

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 62.1%

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

      1. associate-/l* 50.0%

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

      2. associate-/r/ 67.3%

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

   4. Simplified 67.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-291}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 5: 94.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 9.2 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 9.2e-10)))
   (* x (+ (/ y z) -1.0))
   (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 9.2e-10)) {
		tmp = x * ((y / z) + -1.0);
	} 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) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 9.2d-10))) then
        tmp = x * ((y / z) + (-1.0d0))
    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 <= -1.0) || !(y <= 9.2e-10)) {
		tmp = x * ((y / z) + -1.0);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 9.2e-10):
		tmp = x * ((y / z) + -1.0)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 9.2e-10))
		tmp = Float64(x * Float64(Float64(y / z) + -1.0));
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 9.2e-10)))
		tmp = x * ((y / z) + -1.0);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 9.2e-10]], $MachinePrecision]], N[(x * N[(N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 9.20000000000000028e-10 < y

   1. Initial program 82.8%

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

   2. Taylor expanded in x around 0 82.8%

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

      1. associate--l+ 82.8%

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

      2. +-commutative 82.8%

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

      3. associate-*r/ 88.0%

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

      4. +-commutative 88.0%

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

      5. associate--l+ 88.0%

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

      6. div-sub 88.0%

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

      7. sub-neg 88.0%

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

      8. *-inverses 88.0%

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

      9. metadata-eval 88.0%

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

   4. Simplified 88.0%

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

   5. Taylor expanded in y around inf 88.0%

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

   if -1 < y < 9.20000000000000028e-10

   1. Initial program 90.4%

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

   2. Taylor expanded in x around 0 90.4%

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

      1. associate--l+ 90.4%

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

      2. +-commutative 90.4%

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

      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%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. distribute-rgt-in 99.0%

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

      4. associate-*l/ 99.2%

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

      5. *-lft-identity 99.2%

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

      6. neg-mul-1 99.2%

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

      7. unsub-neg 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 9.2 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 6: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.05) (not (<= z 1.0)))
   (* x (+ (/ y z) -1.0))
   (/ (+ x (* x y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05) || !(z <= 1.0)) {
		tmp = x * ((y / z) + -1.0);
	} 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 <= (-1.05d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x * ((y / z) + (-1.0d0))
    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 <= -1.05) || !(z <= 1.0)) {
		tmp = x * ((y / z) + -1.0);
	} else {
		tmp = (x + (x * y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.05) or not (z <= 1.0):
		tmp = x * ((y / z) + -1.0)
	else:
		tmp = (x + (x * y)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.05) || !(z <= 1.0))
		tmp = Float64(x * Float64(Float64(y / z) + -1.0));
	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 <= -1.05) || ~((z <= 1.0)))
		tmp = x * ((y / z) + -1.0);
	else
		tmp = (x + (x * y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.05], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x * N[(N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05000000000000004 or 1 < z

   1. Initial program 70.8%

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

   2. Taylor expanded in x around 0 70.8%

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

      1. associate--l+ 70.8%

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

      2. +-commutative 70.8%

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

      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%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around inf 99.1%

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

   if -1.05000000000000004 < 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 100.0%

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

      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. Taylor expanded in z around 0 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -61000000000 \lor \neg \left(y \leq 1.1 \cdot 10^{+49}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -61000000000.0) (not (<= y 1.1e+49)))
   (* y (/ x z))
   (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -61000000000.0) || !(y <= 1.1e+49)) {
		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 <= (-61000000000.0d0)) .or. (.not. (y <= 1.1d+49))) 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 <= -61000000000.0) || !(y <= 1.1e+49)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -61000000000.0) or not (y <= 1.1e+49):
		tmp = y * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -61000000000.0) || !(y <= 1.1e+49))
		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 <= -61000000000.0) || ~((y <= 1.1e+49)))
		tmp = y * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -61000000000.0], N[Not[LessEqual[y, 1.1e+49]], $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 < -6.1e10 or 1.1e49 < y

   1. Initial program 82.0%

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

   2. Taylor expanded in y around inf 71.8%

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

      1. associate-/l* 65.3%

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

      2. associate-/r/ 74.3%

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

   4. Simplified 74.3%

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

   if -6.1e10 < y < 1.1e49

   1. Initial program 90.2%

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

   2. Taylor expanded in x around 0 90.2%

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

      1. associate--l+ 90.2%

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

      2. +-commutative 90.2%

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

      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%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 97.4%

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

      1. sub-neg 97.4%

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

      2. metadata-eval 97.4%

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

      3. distribute-rgt-in 97.3%

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

      4. associate-*l/ 97.5%

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

      5. *-lft-identity 97.5%

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

      6. neg-mul-1 97.5%

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

      7. unsub-neg 97.5%

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

   7. Simplified 97.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -61000000000 \lor \neg \left(y \leq 1.1 \cdot 10^{+49}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.5e+14)
   (* y (/ x z))
   (if (<= y 2.2e+48) (- (/ x z) x) (/ (* x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.5e+14) {
		tmp = y * (x / z);
	} else if (y <= 2.2e+48) {
		tmp = (x / z) - x;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.5d+14)) then
        tmp = y * (x / z)
    else if (y <= 2.2d+48) then
        tmp = (x / z) - x
    else
        tmp = (x * y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.5e+14) {
		tmp = y * (x / z);
	} else if (y <= 2.2e+48) {
		tmp = (x / z) - x;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.5e+14:
		tmp = y * (x / z)
	elif y <= 2.2e+48:
		tmp = (x / z) - x
	else:
		tmp = (x * y) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.5e+14)
		tmp = Float64(y * Float64(x / z));
	elseif (y <= 2.2e+48)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(Float64(x * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.5e+14)
		tmp = y * (x / z);
	elseif (y <= 2.2e+48)
		tmp = (x / z) - x;
	else
		tmp = (x * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.5e+14], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+48], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5e14

   1. Initial program 79.6%

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

   2. Taylor expanded in y around inf 66.9%

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

      1. associate-/l* 63.2%

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

      2. associate-/r/ 73.4%

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

   4. Simplified 73.4%

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

   if -3.5e14 < y < 2.1999999999999999e48

   1. Initial program 90.2%

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

   2. Taylor expanded in x around 0 90.2%

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

      1. associate--l+ 90.2%

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

      2. +-commutative 90.2%

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

      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%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 97.4%

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

      1. sub-neg 97.4%

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

      2. metadata-eval 97.4%

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

      3. distribute-rgt-in 97.3%

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

      4. associate-*l/ 97.5%

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

      5. *-lft-identity 97.5%

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

      6. neg-mul-1 97.5%

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

      7. unsub-neg 97.5%

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

   7. Simplified 97.5%

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

   if 2.1999999999999999e48 < y

   1. Initial program 84.8%

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

   2. Taylor expanded in y around inf 77.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 9: 64.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		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 <= 1.0)))
		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, 1.0]], $MachinePrecision]], (-x), N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 70.8%

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

   2. Taylor expanded in z around inf 77.3%

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

      1. mul-1-neg 77.3%

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

   4. Simplified 77.3%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 60.0%

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

      1. associate-/l* 60.0%

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

   4. Simplified 60.0%

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

   5. Taylor expanded in z around 0 58.4%

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

4. Final simplification 66.9%

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

Alternative 10: 38.5% 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 86.9%

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

2. Taylor expanded in z around inf 36.3%

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

   1. mul-1-neg 36.3%

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

4. Simplified 36.3%

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

5. Final simplification 36.3%

   \[\leadsto -x \]

Developer target: 99.5% 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 2023308 
(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))