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

Percentage Accurate: 87.9% → 99.4%

Time: 7.3s

Alternatives: 11

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * ((y - z) + 1.0d0)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.9% 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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x_m \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \left(\frac{y + 1}{z} + -1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.5e+54)
    (/ (* x_m (+ (- y z) 1.0)) z)
    (* x_m (+ (/ (+ y 1.0) z) -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 tmp;
	if (x_m <= 1.5e+54) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = x_m * (((y + 1.0) / z) + -1.0);
	}
	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 (x_m <= 1.5d+54) then
        tmp = (x_m * ((y - z) + 1.0d0)) / z
    else
        tmp = x_m * (((y + 1.0d0) / z) + (-1.0d0))
    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 (x_m <= 1.5e+54) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = x_m * (((y + 1.0) / z) + -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):
	tmp = 0
	if x_m <= 1.5e+54:
		tmp = (x_m * ((y - z) + 1.0)) / z
	else:
		tmp = x_m * (((y + 1.0) / z) + -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)
	tmp = 0.0
	if (x_m <= 1.5e+54)
		tmp = Float64(Float64(x_m * Float64(Float64(y - z) + 1.0)) / z);
	else
		tmp = Float64(x_m * Float64(Float64(Float64(y + 1.0) / z) + -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)
	tmp = 0.0;
	if (x_m <= 1.5e+54)
		tmp = (x_m * ((y - z) + 1.0)) / z;
	else
		tmp = x_m * (((y + 1.0) / z) + -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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.5e+54], N[(N[(x$95$m * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(N[(N[(y + 1.0), $MachinePrecision] / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4999999999999999e54

   1. Initial program 93.5%

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

   if 1.4999999999999999e54 < x

   1. Initial program 68.6%

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

   3. Taylor expanded in x around 0 68.6%

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

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

   5. Simplified 99.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y + 1}{z} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y \cdot \frac{x_m}{z}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1250000000000:\\ \;\;\;\;-x_m\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-274}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x_m\\ \end{array} \end{array} \end{array} \]

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 -1250000000000.0)
      (- x_m)
      (if (<= z -4.2e-26)
        t_0
        (if (<= z -5.5e-74)
          (/ x_m z)
          (if (<= z -1.35e-152)
            t_0
            (if (<= z 4.5e-274)
              (/ x_m z)
              (if (<= z 4.4e-125)
                t_0
                (if (<= z 1.2e-36)
                  (/ x_m z)
                  (if (<= z 2.35e+45) 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 <= -1250000000000.0) {
		tmp = -x_m;
	} else if (z <= -4.2e-26) {
		tmp = t_0;
	} else if (z <= -5.5e-74) {
		tmp = x_m / z;
	} else if (z <= -1.35e-152) {
		tmp = t_0;
	} else if (z <= 4.5e-274) {
		tmp = x_m / z;
	} else if (z <= 4.4e-125) {
		tmp = t_0;
	} else if (z <= 1.2e-36) {
		tmp = x_m / z;
	} else if (z <= 2.35e+45) {
		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 <= (-1250000000000.0d0)) then
        tmp = -x_m
    else if (z <= (-4.2d-26)) then
        tmp = t_0
    else if (z <= (-5.5d-74)) then
        tmp = x_m / z
    else if (z <= (-1.35d-152)) then
        tmp = t_0
    else if (z <= 4.5d-274) then
        tmp = x_m / z
    else if (z <= 4.4d-125) then
        tmp = t_0
    else if (z <= 1.2d-36) then
        tmp = x_m / z
    else if (z <= 2.35d+45) 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 <= -1250000000000.0) {
		tmp = -x_m;
	} else if (z <= -4.2e-26) {
		tmp = t_0;
	} else if (z <= -5.5e-74) {
		tmp = x_m / z;
	} else if (z <= -1.35e-152) {
		tmp = t_0;
	} else if (z <= 4.5e-274) {
		tmp = x_m / z;
	} else if (z <= 4.4e-125) {
		tmp = t_0;
	} else if (z <= 1.2e-36) {
		tmp = x_m / z;
	} else if (z <= 2.35e+45) {
		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 <= -1250000000000.0:
		tmp = -x_m
	elif z <= -4.2e-26:
		tmp = t_0
	elif z <= -5.5e-74:
		tmp = x_m / z
	elif z <= -1.35e-152:
		tmp = t_0
	elif z <= 4.5e-274:
		tmp = x_m / z
	elif z <= 4.4e-125:
		tmp = t_0
	elif z <= 1.2e-36:
		tmp = x_m / z
	elif z <= 2.35e+45:
		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 <= -1250000000000.0)
		tmp = Float64(-x_m);
	elseif (z <= -4.2e-26)
		tmp = t_0;
	elseif (z <= -5.5e-74)
		tmp = Float64(x_m / z);
	elseif (z <= -1.35e-152)
		tmp = t_0;
	elseif (z <= 4.5e-274)
		tmp = Float64(x_m / z);
	elseif (z <= 4.4e-125)
		tmp = t_0;
	elseif (z <= 1.2e-36)
		tmp = Float64(x_m / z);
	elseif (z <= 2.35e+45)
		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 <= -1250000000000.0)
		tmp = -x_m;
	elseif (z <= -4.2e-26)
		tmp = t_0;
	elseif (z <= -5.5e-74)
		tmp = x_m / z;
	elseif (z <= -1.35e-152)
		tmp = t_0;
	elseif (z <= 4.5e-274)
		tmp = x_m / z;
	elseif (z <= 4.4e-125)
		tmp = t_0;
	elseif (z <= 1.2e-36)
		tmp = x_m / z;
	elseif (z <= 2.35e+45)
		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, -1250000000000.0], (-x$95$m), If[LessEqual[z, -4.2e-26], t$95$0, If[LessEqual[z, -5.5e-74], N[(x$95$m / z), $MachinePrecision], If[LessEqual[z, -1.35e-152], t$95$0, If[LessEqual[z, 4.5e-274], N[(x$95$m / z), $MachinePrecision], If[LessEqual[z, 4.4e-125], t$95$0, If[LessEqual[z, 1.2e-36], N[(x$95$m / z), $MachinePrecision], If[LessEqual[z, 2.35e+45], t$95$0, (-x$95$m)]]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25e12 or 2.35000000000000001e45 < z

   1. Initial program 74.4%

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

   3. Taylor expanded in z around inf 77.7%

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

      1. mul-1-neg 77.7%

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

   5. Simplified 77.7%

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

   if -1.25e12 < z < -4.20000000000000016e-26 or -5.5000000000000001e-74 < z < -1.34999999999999999e-152 or 4.49999999999999991e-274 < z < 4.3999999999999999e-125 or 1.2e-36 < z < 2.35000000000000001e45

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 68.5%

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

      1. associate-/l* 62.6%

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

      2. associate-/r/ 71.2%

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

   5. Simplified 71.2%

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

   if -4.20000000000000016e-26 < z < -5.5000000000000001e-74 or -1.34999999999999999e-152 < z < 4.49999999999999991e-274 or 4.3999999999999999e-125 < z < 1.2e-36

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 76.5%

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

      1. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

   6. Taylor expanded in z around 0 76.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1250000000000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-26}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-152}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-274}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-125}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.5× speedup

Math

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

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 -1.0) (not (<= y 1.0)))
    (* x_m (+ -1.0 (/ y 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 <= -1.0) || !(y <= 1.0)) {
		tmp = x_m * (-1.0 + (y / 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 <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = x_m * ((-1.0d0) + (y / 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 <= -1.0) || !(y <= 1.0)) {
		tmp = x_m * (-1.0 + (y / 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 <= -1.0) or not (y <= 1.0):
		tmp = x_m * (-1.0 + (y / 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 <= -1.0) || !(y <= 1.0))
		tmp = Float64(x_m * Float64(-1.0 + Float64(y / 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 <= -1.0) || ~((y <= 1.0)))
		tmp = x_m * (-1.0 + (y / 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, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x$95$m * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0 87.2%

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

      1. associate--l+ 87.2%

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

      2. +-commutative 87.2%

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

      3. associate-*r/ 93.9%

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

      4. +-commutative 93.9%

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

      5. associate--l+ 93.9%

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

      6. div-sub 93.9%

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

      7. sub-neg 93.9%

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

      8. *-inverses 93.9%

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

      9. metadata-eval 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in y around inf 92.7%

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

   if -1 < y < 1

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0 89.8%

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

      1. associate--l+ 89.8%

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

      2. +-commutative 89.8%

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

      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%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 98.8%

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

      1. sub-neg 98.8%

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

      2. metadata-eval 98.8%

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

      3. distribute-rgt-in 98.8%

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

      4. associate-*l/ 98.9%

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

      5. *-lft-identity 98.9%

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

      6. neg-mul-1 98.9%

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

      7. unsub-neg 98.9%

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

   8. Simplified 98.9%

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

4. Final simplification 95.8%

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

Alternative 4: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -240000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x_m \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x_m}{z}\\ \end{array} \end{array} \]

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 -240000000.0) (not (<= z 1.0)))
    (* x_m (+ -1.0 (/ y z)))
    (* (+ y 1.0) (/ 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 <= -240000000.0) || !(z <= 1.0)) {
		tmp = x_m * (-1.0 + (y / z));
	} else {
		tmp = (y + 1.0) * (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 <= (-240000000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x_m * ((-1.0d0) + (y / z))
    else
        tmp = (y + 1.0d0) * (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 <= -240000000.0) || !(z <= 1.0)) {
		tmp = x_m * (-1.0 + (y / z));
	} else {
		tmp = (y + 1.0) * (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 <= -240000000.0) or not (z <= 1.0):
		tmp = x_m * (-1.0 + (y / z))
	else:
		tmp = (y + 1.0) * (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 <= -240000000.0) || !(z <= 1.0))
		tmp = Float64(x_m * Float64(-1.0 + Float64(y / z)));
	else
		tmp = Float64(Float64(y + 1.0) * 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 <= -240000000.0) || ~((z <= 1.0)))
		tmp = x_m * (-1.0 + (y / z));
	else
		tmp = (y + 1.0) * (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, -240000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x$95$m * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e8 or 1 < z

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0 77.4%

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

      1. associate--l+ 77.4%

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

      2. +-commutative 77.4%

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

      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%

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

   6. Taylor expanded in y around inf 99.6%

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

   if -2.4e8 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.4%

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

      1. associate-/l* 94.2%

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

      2. associate-/r/ 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -240000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+31} \lor \neg \left(y \leq 1.18 \cdot 10^{+82}\right):\\ \;\;\;\;y \cdot \frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \end{array} \end{array} \]

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 -8e+31) (not (<= y 1.18e+82)))
    (* 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 <= -8e+31) || !(y <= 1.18e+82)) {
		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 <= (-8d+31)) .or. (.not. (y <= 1.18d+82))) 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 <= -8e+31) || !(y <= 1.18e+82)) {
		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 <= -8e+31) or not (y <= 1.18e+82):
		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 <= -8e+31) || !(y <= 1.18e+82))
		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 <= -8e+31) || ~((y <= 1.18e+82)))
		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, -8e+31], N[Not[LessEqual[y, 1.18e+82]], $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 < -7.9999999999999997e31 or 1.1800000000000001e82 < y

   1. Initial program 87.5%

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

   3. Taylor expanded in y around inf 75.5%

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

      1. associate-/l* 75.3%

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

      2. associate-/r/ 77.5%

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

   5. Simplified 77.5%

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

   if -7.9999999999999997e31 < y < 1.1800000000000001e82

   1. Initial program 89.2%

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

   3. Taylor expanded in x around 0 89.2%

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

      1. associate--l+ 89.2%

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

      2. +-commutative 89.2%

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

      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%

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

   6. Taylor expanded in y around 0 95.6%

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

      1. sub-neg 95.6%

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

      2. metadata-eval 95.6%

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

      3. distribute-rgt-in 95.6%

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

      4. associate-*l/ 95.7%

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

      5. *-lft-identity 95.7%

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

      6. neg-mul-1 95.7%

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

      7. unsub-neg 95.7%

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

   8. Simplified 95.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+31} \lor \neg \left(y \leq 1.18 \cdot 10^{+82}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \frac{x_m}{z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+71}:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{\frac{z}{y}}\\ \end{array} \end{array} \]

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 -1.4e+31)
    (* y (/ x_m z))
    (if (<= y 1.3e+71) (- (/ 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 <= -1.4e+31) {
		tmp = y * (x_m / z);
	} else if (y <= 1.3e+71) {
		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 <= (-1.4d+31)) then
        tmp = y * (x_m / z)
    else if (y <= 1.3d+71) 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 <= -1.4e+31) {
		tmp = y * (x_m / z);
	} else if (y <= 1.3e+71) {
		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 <= -1.4e+31:
		tmp = y * (x_m / z)
	elif y <= 1.3e+71:
		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 <= -1.4e+31)
		tmp = Float64(y * Float64(x_m / z));
	elseif (y <= 1.3e+71)
		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 <= -1.4e+31)
		tmp = y * (x_m / z);
	elseif (y <= 1.3e+71)
		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, -1.4e+31], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+71], 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 < -1.40000000000000008e31

   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 76.9%

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

      1. associate-/l* 75.7%

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

      2. associate-/r/ 79.4%

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

   5. Simplified 79.4%

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

   if -1.40000000000000008e31 < y < 1.29999999999999996e71

   1. Initial program 89.2%

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

   3. Taylor expanded in x around 0 89.2%

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

      1. associate--l+ 89.2%

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

      2. +-commutative 89.2%

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

      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%

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

   6. Taylor expanded in y around 0 95.6%

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

      1. sub-neg 95.6%

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

      2. metadata-eval 95.6%

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

      3. distribute-rgt-in 95.6%

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

      4. associate-*l/ 95.7%

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

      5. *-lft-identity 95.7%

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

      6. neg-mul-1 95.7%

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

      7. unsub-neg 95.7%

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

   8. Simplified 95.7%

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

   if 1.29999999999999996e71 < y

   1. Initial program 86.6%

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

   3. Taylor expanded in y around inf 72.7%

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

      1. associate-/l* 74.4%

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

   5. Simplified 74.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+32}:\\ \;\;\;\;\frac{y}{\frac{z}{x_m}}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{\frac{z}{y}}\\ \end{array} \end{array} \]

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 -1.1e+32)
    (/ y (/ z x_m))
    (if (<= y 6.6e+56) (- (/ 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 <= -1.1e+32) {
		tmp = y / (z / x_m);
	} else if (y <= 6.6e+56) {
		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 <= (-1.1d+32)) then
        tmp = y / (z / x_m)
    else if (y <= 6.6d+56) 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 <= -1.1e+32) {
		tmp = y / (z / x_m);
	} else if (y <= 6.6e+56) {
		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 <= -1.1e+32:
		tmp = y / (z / x_m)
	elif y <= 6.6e+56:
		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 <= -1.1e+32)
		tmp = Float64(y / Float64(z / x_m));
	elseif (y <= 6.6e+56)
		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 <= -1.1e+32)
		tmp = y / (z / x_m);
	elseif (y <= 6.6e+56)
		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, -1.1e+32], N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.6e+56], 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 < -1.1e32

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0 88.0%

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

      1. associate--l+ 88.0%

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

      2. +-commutative 88.0%

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

      3. associate-*r/ 93.1%

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

      4. +-commutative 93.1%

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

      5. associate--l+ 93.1%

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

      6. div-sub 93.1%

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

      7. sub-neg 93.1%

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

      8. *-inverses 93.1%

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

      9. metadata-eval 93.1%

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

   5. Simplified 93.1%

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

   6. Taylor expanded in y around inf 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-/l* 79.5%

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

   8. Simplified 79.5%

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

   if -1.1e32 < y < 6.60000000000000004e56

   1. Initial program 89.2%

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

   3. Taylor expanded in x around 0 89.2%

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

      1. associate--l+ 89.2%

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

      2. +-commutative 89.2%

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

      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%

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

   6. Taylor expanded in y around 0 95.6%

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

      1. sub-neg 95.6%

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

      2. metadata-eval 95.6%

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

      3. distribute-rgt-in 95.6%

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

      4. associate-*l/ 95.7%

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

      5. *-lft-identity 95.7%

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

      6. neg-mul-1 95.7%

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

      7. unsub-neg 95.7%

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

   8. Simplified 95.7%

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

   if 6.60000000000000004e56 < y

   1. Initial program 86.6%

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

   3. Taylor expanded in y around inf 72.7%

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

      1. associate-/l* 74.4%

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

   5. Simplified 74.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+32}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-16} \lor \neg \left(z \leq 880000\right):\\ \;\;\;\;-x_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z}\\ \end{array} \end{array} \]

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 -3.8e-16) (not (<= z 880000.0))) (- 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 <= -3.8e-16) || !(z <= 880000.0)) {
		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 <= (-3.8d-16)) .or. (.not. (z <= 880000.0d0))) 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 <= -3.8e-16) || !(z <= 880000.0)) {
		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 <= -3.8e-16) or not (z <= 880000.0):
		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 <= -3.8e-16) || !(z <= 880000.0))
		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 <= -3.8e-16) || ~((z <= 880000.0)))
		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, -3.8e-16], N[Not[LessEqual[z, 880000.0]], $MachinePrecision]], (-x$95$m), N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.80000000000000012e-16 or 8.8e5 < z

   1. Initial program 78.4%

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

   3. Taylor expanded in z around inf 68.7%

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

      1. mul-1-neg 68.7%

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

   5. Simplified 68.7%

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

   if -3.80000000000000012e-16 < z < 8.8e5

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 62.4%

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

      1. associate-/l* 62.4%

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

   5. Simplified 62.4%

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

   6. Taylor expanded in z around 0 62.0%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-16} \lor \neg \left(z \leq 880000\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.0% accurate, 1.0× speedup

Math

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

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 (+ (/ (+ y 1.0) z) -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) {
	return x_s * (x_m * (((y + 1.0) / z) + -1.0));
}

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 * (((y + 1.0d0) / z) + (-1.0d0)))
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 * (((y + 1.0) / z) + -1.0));
}

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

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 * Float64(Float64(Float64(y + 1.0) / z) + -1.0)))
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 * (((y + 1.0) / z) + -1.0));
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 * N[(x$95$m * N[(N[(N[(y + 1.0), $MachinePrecision] / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in x around 0 88.5%

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

   1. associate--l+ 88.5%

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

   2. +-commutative 88.5%

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

   3. associate-*r/ 96.9%

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

   4. +-commutative 96.9%

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

   5. associate--l+ 96.9%

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

   6. div-sub 96.9%

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

   7. sub-neg 96.9%

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

   8. *-inverses 96.9%

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

   9. metadata-eval 96.9%

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

5. Simplified 96.9%

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

6. Final simplification 96.9%

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

Alternative 10: 39.0% 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 88.5%

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

3. Taylor expanded in z around inf 37.9%

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

   1. mul-1-neg 37.9%

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

5. Simplified 37.9%

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

6. Final simplification 37.9%

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

Alternative 11: 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 88.5%

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

3. Taylor expanded in z around inf 29.1%

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

   1. associate-*r* 29.1%

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

   2. mul-1-neg 29.1%

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

5. Simplified 29.1%

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

   1. div-inv 29.1%

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

   2. associate-*l* 37.9%

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

   3. div-inv 37.9%

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

   4. *-inverses 37.9%

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

   5. *-commutative 37.9%

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

   6. *-un-lft-identity 37.9%

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

   7. neg-sub0 37.9%

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

   8. sub-neg 37.9%

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

   9. add-sqr-sqrt 18.7%

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

   10. sqrt-unprod 18.7%

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

   11. sqr-neg 18.7%

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

   12. sqrt-unprod 1.4%

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

   13. add-sqr-sqrt 2.9%

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

7. Applied egg-rr 2.9%

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

   1. +-lft-identity 2.9%

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

9. Simplified 2.9%

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

10. Final simplification 2.9%

    \[\leadsto x \]
11. Add Preprocessing

Developer target: 99.3% 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 2024019 
(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))