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

Percentage Accurate: 87.5% → 93.3%

Time: 8.4s

Alternatives: 8

Speedup: 0.3×

• 20.3% 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: 87.5% 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: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x_m \cdot \left(\left(y - z\right) + 1\right)}{z}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x_m}}{y + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (+ (- y z) 1.0)) z)))
   (*
    x_s
    (if (<= t_0 (- INFINITY))
      (- (/ x_m z) x_m)
      (if (<= t_0 5e+305) t_0 (/ 1.0 (/ (/ z x_m) (+ y 1.0))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * ((y - z) + 1.0)) / z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x_m / z) - x_m;
	} else if (t_0 <= 5e+305) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((z / x_m) / (y + 1.0));
	}
	return x_s * tmp;
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * ((y - z) + 1.0)) / z;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x_m / z) - x_m;
	} else if (t_0 <= 5e+305) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((z / x_m) / (y + 1.0));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * ((y - z) + 1.0)) / z
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x_m / z) - x_m
	elif t_0 <= 5e+305:
		tmp = t_0
	else:
		tmp = 1.0 / ((z / x_m) / (y + 1.0))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(Float64(y - z) + 1.0)) / z)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x_m / z) - x_m);
	elseif (t_0 <= 5e+305)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(z / x_m) / Float64(y + 1.0)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * ((y - z) + 1.0)) / z;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x_m / z) - x_m;
	elseif (t_0 <= 5e+305)
		tmp = t_0;
	else
		tmp = 1.0 / ((z / x_m) / (y + 1.0));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e+305], t$95$0, N[(1.0 / N[(N[(z / x$95$m), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -inf.0

   1. Initial program 65.5%

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

   3. Taylor expanded in y around 0 40.2%

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

   4. Taylor expanded in z around 0 69.5%

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

      1. neg-mul-1 69.5%

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

      2. +-commutative 69.5%

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

      3. unsub-neg 69.5%

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

   6. Simplified 69.5%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 5.00000000000000009e305

   1. Initial program 99.8%

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

   if 5.00000000000000009e305 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 63.2%

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

   3. Taylor expanded in z around 0 61.5%

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

      1. clear-num 61.5%

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

      2. inv-pow 61.5%

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

      3. +-commutative 61.5%

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

   5. Applied egg-rr 61.5%

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

      1. unpow-1 61.5%

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

      2. associate-/r* 68.9%

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

   7. Simplified 68.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -\infty:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y \cdot \frac{x_m}{z}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x_m\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-203}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x_m\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* y (/ x_m z))))
   (*
    x_s
    (if (<= z -1.0)
      (- x_m)
      (if (<= z 7.2e-203)
        (/ x_m z)
        (if (<= z 4.2e-177)
          t_0
          (if (<= z 3.7e-81) (/ x_m z) (if (<= z 3.15e+38) t_0 (- x_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (x_m / z);
	double tmp;
	if (z <= -1.0) {
		tmp = -x_m;
	} else if (z <= 7.2e-203) {
		tmp = x_m / z;
	} else if (z <= 4.2e-177) {
		tmp = t_0;
	} else if (z <= 3.7e-81) {
		tmp = x_m / z;
	} else if (z <= 3.15e+38) {
		tmp = t_0;
	} else {
		tmp = -x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x_m / z)
    if (z <= (-1.0d0)) then
        tmp = -x_m
    else if (z <= 7.2d-203) then
        tmp = x_m / z
    else if (z <= 4.2d-177) then
        tmp = t_0
    else if (z <= 3.7d-81) then
        tmp = x_m / z
    else if (z <= 3.15d+38) then
        tmp = t_0
    else
        tmp = -x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (x_m / z);
	double tmp;
	if (z <= -1.0) {
		tmp = -x_m;
	} else if (z <= 7.2e-203) {
		tmp = x_m / z;
	} else if (z <= 4.2e-177) {
		tmp = t_0;
	} else if (z <= 3.7e-81) {
		tmp = x_m / z;
	} else if (z <= 3.15e+38) {
		tmp = t_0;
	} else {
		tmp = -x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = y * (x_m / z)
	tmp = 0
	if z <= -1.0:
		tmp = -x_m
	elif z <= 7.2e-203:
		tmp = x_m / z
	elif z <= 4.2e-177:
		tmp = t_0
	elif z <= 3.7e-81:
		tmp = x_m / z
	elif z <= 3.15e+38:
		tmp = t_0
	else:
		tmp = -x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(y * Float64(x_m / z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(-x_m);
	elseif (z <= 7.2e-203)
		tmp = Float64(x_m / z);
	elseif (z <= 4.2e-177)
		tmp = t_0;
	elseif (z <= 3.7e-81)
		tmp = Float64(x_m / z);
	elseif (z <= 3.15e+38)
		tmp = t_0;
	else
		tmp = Float64(-x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = y * (x_m / z);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = -x_m;
	elseif (z <= 7.2e-203)
		tmp = x_m / z;
	elseif (z <= 4.2e-177)
		tmp = t_0;
	elseif (z <= 3.7e-81)
		tmp = x_m / z;
	elseif (z <= 3.15e+38)
		tmp = t_0;
	else
		tmp = -x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.0], (-x$95$m), If[LessEqual[z, 7.2e-203], N[(x$95$m / z), $MachinePrecision], If[LessEqual[z, 4.2e-177], t$95$0, If[LessEqual[z, 3.7e-81], N[(x$95$m / z), $MachinePrecision], If[LessEqual[z, 3.15e+38], t$95$0, (-x$95$m)]]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 3.15000000000000001e38 < z

   1. Initial program 73.8%

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

   3. Taylor expanded in z around inf 71.7%

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

      1. mul-1-neg 71.7%

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

   5. Simplified 71.7%

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

   if -1 < z < 7.19999999999999958e-203 or 4.20000000000000002e-177 < z < 3.69999999999999986e-81

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 69.2%

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

   4. Taylor expanded in z around 0 68.5%

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

   if 7.19999999999999958e-203 < z < 4.20000000000000002e-177 or 3.69999999999999986e-81 < z < 3.15000000000000001e38

   1. Initial program 97.1%

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

   3. Taylor expanded in y around inf 66.4%

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

      1. associate-/l* 59.5%

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

      2. associate-/r/ 74.2%

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

   5. Simplified 74.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-203}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-177}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{+38}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+29} \lor \neg \left(y \leq 6.6 \cdot 10^{+35}\right):\\ \;\;\;\;y \cdot \frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -4.8e+29) (not (<= y 6.6e+35)))
    (* y (/ x_m z))
    (- (/ x_m z) x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -4.8e+29) || !(y <= 6.6e+35)) {
		tmp = y * (x_m / z);
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-4.8d+29)) .or. (.not. (y <= 6.6d+35))) then
        tmp = y * (x_m / z)
    else
        tmp = (x_m / z) - x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -4.8e+29) || !(y <= 6.6e+35)) {
		tmp = y * (x_m / z);
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (y <= -4.8e+29) or not (y <= 6.6e+35):
		tmp = y * (x_m / z)
	else:
		tmp = (x_m / z) - x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((y <= -4.8e+29) || !(y <= 6.6e+35))
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / z) - x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y <= -4.8e+29) || ~((y <= 6.6e+35)))
		tmp = y * (x_m / z);
	else
		tmp = (x_m / z) - x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[y, -4.8e+29], N[Not[LessEqual[y, 6.6e+35]], $MachinePrecision]], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000002e29 or 6.6000000000000003e35 < y

   1. Initial program 88.0%

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

   3. Taylor expanded in y around inf 76.8%

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

      1. associate-/l* 76.6%

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

      2. associate-/r/ 75.9%

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

   5. Simplified 75.9%

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

   if -4.8000000000000002e29 < y < 6.6000000000000003e35

   1. Initial program 87.1%

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

   3. Taylor expanded in y around 0 83.3%

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

   4. Taylor expanded in z around 0 96.1%

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

      1. neg-mul-1 96.1%

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

      2. +-commutative 96.1%

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

      3. unsub-neg 96.1%

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

   6. Simplified 96.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+29} \lor \neg \left(y \leq 6.6 \cdot 10^{+35}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+29} \lor \neg \left(y \leq 9.5 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{x_m}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -3.8e+29) (not (<= y 9.5e+35)))
    (/ x_m (/ z y))
    (- (/ x_m z) x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -3.8e+29) || !(y <= 9.5e+35)) {
		tmp = x_m / (z / y);
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.8d+29)) .or. (.not. (y <= 9.5d+35))) then
        tmp = x_m / (z / y)
    else
        tmp = (x_m / z) - x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -3.8e+29) || !(y <= 9.5e+35)) {
		tmp = x_m / (z / y);
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (y <= -3.8e+29) or not (y <= 9.5e+35):
		tmp = x_m / (z / y)
	else:
		tmp = (x_m / z) - x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((y <= -3.8e+29) || !(y <= 9.5e+35))
		tmp = Float64(x_m / Float64(z / y));
	else
		tmp = Float64(Float64(x_m / z) - x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y <= -3.8e+29) || ~((y <= 9.5e+35)))
		tmp = x_m / (z / y);
	else
		tmp = (x_m / z) - x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[y, -3.8e+29], N[Not[LessEqual[y, 9.5e+35]], $MachinePrecision]], N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -3.79999999999999971e29 or 9.50000000000000062e35 < y

   1. Initial program 88.0%

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

   3. Taylor expanded in y around inf 76.8%

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

      1. associate-/l* 76.6%

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

   5. Simplified 76.6%

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

   if -3.79999999999999971e29 < y < 9.50000000000000062e35

   1. Initial program 87.1%

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

   3. Taylor expanded in y around 0 83.3%

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

   4. Taylor expanded in z around 0 96.1%

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

      1. neg-mul-1 96.1%

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

      2. +-commutative 96.1%

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

      3. unsub-neg 96.1%

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

   6. Simplified 96.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+29} \lor \neg \left(y \leq 9.5 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x_m \cdot y}{z}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -3.2e+29)
    (/ (* x_m y) z)
    (if (<= y 7.2e+35) (- (/ x_m z) x_m) (/ x_m (/ z y))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -3.2e+29) {
		tmp = (x_m * y) / z;
	} else if (y <= 7.2e+35) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = x_m / (z / y);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.2d+29)) then
        tmp = (x_m * y) / z
    else if (y <= 7.2d+35) then
        tmp = (x_m / z) - x_m
    else
        tmp = x_m / (z / y)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -3.2e+29) {
		tmp = (x_m * y) / z;
	} else if (y <= 7.2e+35) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = x_m / (z / y);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -3.2e+29:
		tmp = (x_m * y) / z
	elif y <= 7.2e+35:
		tmp = (x_m / z) - x_m
	else:
		tmp = x_m / (z / y)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -3.2e+29)
		tmp = Float64(Float64(x_m * y) / z);
	elseif (y <= 7.2e+35)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(x_m / Float64(z / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -3.2e+29)
		tmp = (x_m * y) / z;
	elseif (y <= 7.2e+35)
		tmp = (x_m / z) - x_m;
	else
		tmp = x_m / (z / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -3.2e+29], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 7.2e+35], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.19999999999999987e29

   1. Initial program 86.7%

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

   3. Taylor expanded in y around inf 73.7%

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

   if -3.19999999999999987e29 < y < 7.2000000000000001e35

   1. Initial program 87.1%

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

   3. Taylor expanded in y around 0 83.3%

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

   4. Taylor expanded in z around 0 96.1%

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

      1. neg-mul-1 96.1%

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

      2. +-commutative 96.1%

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

      3. unsub-neg 96.1%

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

   6. Simplified 96.1%

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

   if 7.2000000000000001e35 < y

   1. Initial program 89.5%

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

   3. Taylor expanded in y around inf 80.5%

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

      1. associate-/l* 82.6%

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

   5. Simplified 82.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.205\right):\\ \;\;\;\;-x_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (or (<= z -1.0) (not (<= z 0.205))) (- x_m) (/ x_m z))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.205)) {
		tmp = -x_m;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.205d0))) then
        tmp = -x_m
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.205)) {
		tmp = -x_m;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.205):
		tmp = -x_m
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.205))
		tmp = Float64(-x_m);
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.205)))
		tmp = -x_m;
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.205]], $MachinePrecision]], (-x$95$m), N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.204999999999999988 < z

   1. Initial program 74.5%

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

   3. Taylor expanded in z around inf 68.6%

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

      1. mul-1-neg 68.6%

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

   5. Simplified 68.6%

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

   if -1 < z < 0.204999999999999988

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 61.8%

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

   4. Taylor expanded in z around 0 60.9%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.205\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(-x_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (- x_m)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * -x_m;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * -x_m
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * -x_m;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * -x_m

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(-x_m))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * -x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * (-x$95$m)), $MachinePrecision]

Derivation

1. Initial program 87.5%

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

3. Taylor expanded in z around inf 35.2%

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

   1. mul-1-neg 35.2%

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

5. Simplified 35.2%

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

6. Final simplification 35.2%

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

Alternative 8: 3.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot x_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s x_m))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * x_m
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * x_m

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * x_m)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 87.5%

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

3. Taylor expanded in z around inf 24.9%

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

   1. associate-*r* 24.9%

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

   2. mul-1-neg 24.9%

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

5. Simplified 24.9%

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

   1. div-inv 24.9%

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

   2. associate-*l* 35.1%

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

   3. div-inv 35.2%

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

   4. *-inverses 35.2%

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

   5. *-commutative 35.2%

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

   6. neg-sub0 35.2%

      \[\leadsto 1 \cdot \color{blue}{\left(0 - x\right)} \]

   7. *-un-lft-identity 35.2%

      \[\leadsto \color{blue}{0 - x} \]

   8. sub-neg 35.2%

      \[\leadsto \color{blue}{0 + \left(-x\right)} \]

   9. add-sqr-sqrt 18.6%

      \[\leadsto 0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}} \]

   10. sqrt-unprod 16.9%

       \[\leadsto 0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \]

   11. sqr-neg 16.9%

       \[\leadsto 0 + \sqrt{\color{blue}{x \cdot x}} \]

   12. sqrt-unprod 1.5%

       \[\leadsto 0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}} \]

   13. add-sqr-sqrt 3.0%

       \[\leadsto 0 + \color{blue}{x} \]

7. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{0 + x} \]
8. Step-by-step derivation

   1. +-lft-identity 3.0%

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

9. Simplified 3.0%

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

10. Final simplification 3.0%

    \[\leadsto x \]
11. Add Preprocessing

Developer target: 99.2% 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 2024020 
(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))