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

Percentage Accurate: 87.8% → 99.7%

Time: 6.8s

Alternatives: 10

Speedup: 0.8×

• 20.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

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

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 6.6e-32)
    (/ (fma x_m (- y z) x_m) z)
    (* (/ x_m z) (+ (- y 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 <= 6.6e-32) {
		tmp = fma(x_m, (y - z), x_m) / z;
	} else {
		tmp = (x_m / z) * ((y - 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 <= 6.6e-32)
		tmp = Float64(fma(x_m, Float64(y - z), x_m) / z);
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y - z) + 1.0));
	end
	return Float64(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, 6.6e-32], N[(N[(x$95$m * N[(y - z), $MachinePrecision] + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6.60000000000000051e-32

   1. Initial program 91.2%

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

      1. distribute-lft-in 91.2%

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

      2. fma-def 91.2%

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

      3. *-rgt-identity 91.2%

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

   3. Simplified 91.2%

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

   if 6.60000000000000051e-32 < x

   1. Initial program 83.1%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array} \]

Alternative 2: 65.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y \cdot \frac{x_m}{z}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3800:\\ \;\;\;\;-x_m\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-144}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;-x_m\\ \end{array} \end{array} \end{array} \]

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 -3800.0)
      (- x_m)
      (if (<= z -1.8e-91)
        t_0
        (if (<= z 2.7e-144)
          (/ x_m z)
          (if (<= z 1.3e-75) t_0 (if (<= z 1.0) (/ 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 t_0 = y * (x_m / z);
	double tmp;
	if (z <= -3800.0) {
		tmp = -x_m;
	} else if (z <= -1.8e-91) {
		tmp = t_0;
	} else if (z <= 2.7e-144) {
		tmp = x_m / z;
	} else if (z <= 1.3e-75) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x_m / z;
	} 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 <= (-3800.0d0)) then
        tmp = -x_m
    else if (z <= (-1.8d-91)) then
        tmp = t_0
    else if (z <= 2.7d-144) then
        tmp = x_m / z
    else if (z <= 1.3d-75) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = x_m / z
    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 <= -3800.0) {
		tmp = -x_m;
	} else if (z <= -1.8e-91) {
		tmp = t_0;
	} else if (z <= 2.7e-144) {
		tmp = x_m / z;
	} else if (z <= 1.3e-75) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x_m / z;
	} 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 <= -3800.0:
		tmp = -x_m
	elif z <= -1.8e-91:
		tmp = t_0
	elif z <= 2.7e-144:
		tmp = x_m / z
	elif z <= 1.3e-75:
		tmp = t_0
	elif z <= 1.0:
		tmp = x_m / z
	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 <= -3800.0)
		tmp = Float64(-x_m);
	elseif (z <= -1.8e-91)
		tmp = t_0;
	elseif (z <= 2.7e-144)
		tmp = Float64(x_m / z);
	elseif (z <= 1.3e-75)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(x_m / z);
	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 <= -3800.0)
		tmp = -x_m;
	elseif (z <= -1.8e-91)
		tmp = t_0;
	elseif (z <= 2.7e-144)
		tmp = x_m / z;
	elseif (z <= 1.3e-75)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = x_m / z;
	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, -3800.0], (-x$95$m), If[LessEqual[z, -1.8e-91], t$95$0, If[LessEqual[z, 2.7e-144], N[(x$95$m / z), $MachinePrecision], If[LessEqual[z, 1.3e-75], t$95$0, If[LessEqual[z, 1.0], N[(x$95$m / z), $MachinePrecision], (-x$95$m)]]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -3800 or 1 < z

   1. Initial program 77.1%

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

   2. Taylor expanded in z around inf 82.2%

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

      1. mul-1-neg 82.2%

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

   4. Simplified 82.2%

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

   if -3800 < z < -1.8e-91 or 2.69999999999999975e-144 < z < 1.3e-75

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 64.9%

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

      1. associate-/l* 63.7%

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

      2. associate-/r/ 70.2%

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

   4. Applied egg-rr 70.2%

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

   if -1.8e-91 < z < 2.69999999999999975e-144 or 1.3e-75 < z < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 99.4%

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

      1. associate-/l* 93.1%

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

   4. Simplified 93.1%

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

   5. Taylor expanded in y around 0 65.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3800:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-144}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-75}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+206} \lor \neg \left(z \leq 6.5 \cdot 10^{+167}\right):\\ \;\;\;\;-x_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -9.5e+206) (not (<= z 6.5e+167)))
    (- x_m)
    (* (/ x_m z) (+ (- y 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 ((z <= -9.5e+206) || !(z <= 6.5e+167)) {
		tmp = -x_m;
	} else {
		tmp = (x_m / z) * ((y - 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 ((z <= (-9.5d+206)) .or. (.not. (z <= 6.5d+167))) then
        tmp = -x_m
    else
        tmp = (x_m / z) * ((y - 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 ((z <= -9.5e+206) || !(z <= 6.5e+167)) {
		tmp = -x_m;
	} else {
		tmp = (x_m / z) * ((y - 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 (z <= -9.5e+206) or not (z <= 6.5e+167):
		tmp = -x_m
	else:
		tmp = (x_m / z) * ((y - 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 ((z <= -9.5e+206) || !(z <= 6.5e+167))
		tmp = Float64(-x_m);
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y - 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 ((z <= -9.5e+206) || ~((z <= 6.5e+167)))
		tmp = -x_m;
	else
		tmp = (x_m / z) * ((y - 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[Or[LessEqual[z, -9.5e+206], N[Not[LessEqual[z, 6.5e+167]], $MachinePrecision]], (-x$95$m), N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -9.49999999999999966e206 or 6.5e167 < z

   1. Initial program 63.9%

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

   2. Taylor expanded in z around inf 95.3%

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

      1. mul-1-neg 95.3%

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

   4. Simplified 95.3%

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

   if -9.49999999999999966e206 < z < 6.5e167

   1. Initial program 95.8%

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

      1. associate-/l* 96.6%

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

      2. associate-/r/ 96.1%

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

   3. Applied egg-rr 96.1%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+206} \lor \neg \left(z \leq 6.5 \cdot 10^{+167}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array} \]

Alternative 4: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.15 \lor \neg \left(z \leq 1.12 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m \cdot \left(y + 1\right)}{z}\\ \end{array} \end{array} \]

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 -1.15) (not (<= z 1.12e-14)))
    (- (/ x_m z) x_m)
    (/ (* x_m (+ y 1.0)) 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 <= -1.15) || !(z <= 1.12e-14)) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = (x_m * (y + 1.0)) / 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 <= (-1.15d0)) .or. (.not. (z <= 1.12d-14))) then
        tmp = (x_m / z) - x_m
    else
        tmp = (x_m * (y + 1.0d0)) / 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 <= -1.15) || !(z <= 1.12e-14)) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = (x_m * (y + 1.0)) / 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 <= -1.15) or not (z <= 1.12e-14):
		tmp = (x_m / z) - x_m
	else:
		tmp = (x_m * (y + 1.0)) / 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 <= -1.15) || !(z <= 1.12e-14))
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(Float64(x_m * Float64(y + 1.0)) / 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 <= -1.15) || ~((z <= 1.12e-14)))
		tmp = (x_m / z) - x_m;
	else
		tmp = (x_m * (y + 1.0)) / 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, -1.15], N[Not[LessEqual[z, 1.12e-14]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(x$95$m * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.1499999999999999 or 1.12000000000000006e-14 < z

   1. Initial program 77.4%

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

      1. distribute-lft-in 77.4%

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

      2. *-rgt-identity 77.4%

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

   3. Applied egg-rr 77.4%

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

      1. add-cbrt-cube 41.8%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x \cdot \left(y - z\right)\right) \cdot \left(x \cdot \left(y - z\right)\right)\right) \cdot \left(x \cdot \left(y - z\right)\right)}} + x}{z} \]

      2. pow3 41.8%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(x \cdot \left(y - z\right)\right)}^{3}}} + x}{z} \]

   5. Applied egg-rr 41.8%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(x \cdot \left(y - z\right)\right)}^{3}}} + x}{z} \]

   6. Taylor expanded in y around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unsub-neg 64.2%

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

   8. Simplified 64.2%

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

   9. Taylor expanded in x around 0 64.2%

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

       1. *-lft-identity 64.2%

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

       2. associate-*l/ 64.1%

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

       3. associate-*r* 61.2%

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

       4. sub-neg 61.2%

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

       5. distribute-rgt-in 61.2%

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

       6. associate-*l/ 61.2%

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

       7. *-lft-identity 61.2%

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

       8. *-lft-identity 61.2%

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

       9. associate-*l/ 61.4%

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

       10. *-lft-identity 61.4%

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

       11. cancel-sign-sub-inv 61.4%

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

       12. *-commutative 61.4%

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

       13. associate-/r/ 83.2%

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

       14. *-inverses 83.2%

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

       15. /-rgt-identity 83.2%

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

   11. Simplified 83.2%

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

   if -1.1499999999999999 < z < 1.12000000000000006e-14

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.9%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \lor \neg \left(z \leq 1.12 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \]

Alternative 5: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(y - z\right) + 1\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x_m \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z} \cdot t_0\\ \end{array} \end{array} \end{array} \]

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 z) 1.0)))
   (* x_s (if (<= x_m 4e-32) (/ (* x_m t_0) z) (* (/ x_m z) t_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 = (y - z) + 1.0;
	double tmp;
	if (x_m <= 4e-32) {
		tmp = (x_m * t_0) / z;
	} else {
		tmp = (x_m / z) * t_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) :: t_0
    real(8) :: tmp
    t_0 = (y - z) + 1.0d0
    if (x_m <= 4d-32) then
        tmp = (x_m * t_0) / z
    else
        tmp = (x_m / z) * t_0
    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 - z) + 1.0;
	double tmp;
	if (x_m <= 4e-32) {
		tmp = (x_m * t_0) / z;
	} else {
		tmp = (x_m / z) * t_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 = (y - z) + 1.0
	tmp = 0
	if x_m <= 4e-32:
		tmp = (x_m * t_0) / z
	else:
		tmp = (x_m / z) * t_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(y - z) + 1.0)
	tmp = 0.0
	if (x_m <= 4e-32)
		tmp = Float64(Float64(x_m * t_0) / z);
	else
		tmp = Float64(Float64(x_m / z) * t_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 = (y - z) + 1.0;
	tmp = 0.0;
	if (x_m <= 4e-32)
		tmp = (x_m * t_0) / z;
	else
		tmp = (x_m / z) * t_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[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 4e-32], N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000022e-32

   1. Initial program 91.2%

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

   if 4.00000000000000022e-32 < x

   1. Initial program 83.3%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array} \]

Alternative 6: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+19} \lor \neg \left(y \leq 2.2 \cdot 10^{+14}\right):\\ \;\;\;\;y \cdot \frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \end{array} \end{array} \]

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 -1.25e+19) (not (<= y 2.2e+14)))
    (* 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 <= -1.25e+19) || !(y <= 2.2e+14)) {
		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 <= (-1.25d+19)) .or. (.not. (y <= 2.2d+14))) 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 <= -1.25e+19) || !(y <= 2.2e+14)) {
		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 <= -1.25e+19) or not (y <= 2.2e+14):
		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 <= -1.25e+19) || !(y <= 2.2e+14))
		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 <= -1.25e+19) || ~((y <= 2.2e+14)))
		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, -1.25e+19], N[Not[LessEqual[y, 2.2e+14]], $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 < -1.25e19 or 2.2e14 < y

   1. Initial program 86.3%

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

   2. Taylor expanded in y around inf 76.3%

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

      1. associate-/l* 73.2%

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

      2. associate-/r/ 78.0%

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

   4. Applied egg-rr 78.0%

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

   if -1.25e19 < y < 2.2e14

   1. Initial program 90.5%

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

      1. distribute-lft-in 90.5%

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

      2. *-rgt-identity 90.5%

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

   3. Applied egg-rr 90.5%

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

      1. add-cbrt-cube 60.4%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x \cdot \left(y - z\right)\right) \cdot \left(x \cdot \left(y - z\right)\right)\right) \cdot \left(x \cdot \left(y - z\right)\right)}} + x}{z} \]

      2. pow3 60.4%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(x \cdot \left(y - z\right)\right)}^{3}}} + x}{z} \]

   5. Applied egg-rr 60.4%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(x \cdot \left(y - z\right)\right)}^{3}}} + x}{z} \]

   6. Taylor expanded in y around 0 87.2%

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

      1. mul-1-neg 87.2%

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

      2. unsub-neg 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in x around 0 87.2%

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

       1. *-lft-identity 87.2%

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

       2. associate-*l/ 86.9%

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

       3. associate-*r* 82.7%

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

       4. sub-neg 82.7%

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

       5. distribute-rgt-in 74.6%

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

       6. associate-*l/ 74.8%

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

       7. *-lft-identity 74.8%

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

       8. *-lft-identity 74.8%

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

       9. associate-*l/ 74.9%

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

       10. *-lft-identity 74.9%

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

       11. cancel-sign-sub-inv 74.9%

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

       12. *-commutative 74.9%

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

       13. associate-/r/ 96.6%

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

       14. *-inverses 96.6%

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

       15. /-rgt-identity 96.6%

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

   11. Simplified 96.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+19} \lor \neg \left(y \leq 2.2 \cdot 10^{+14}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 7: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \frac{x_m}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -7.2e+19)
    (* y (/ x_m z))
    (if (<= y 2.1e+14) (- (/ x_m z) x_m) (/ y (/ 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 <= -7.2e+19) {
		tmp = y * (x_m / z);
	} else if (y <= 2.1e+14) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = y / (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 <= (-7.2d+19)) then
        tmp = y * (x_m / z)
    else if (y <= 2.1d+14) then
        tmp = (x_m / z) - x_m
    else
        tmp = y / (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 <= -7.2e+19) {
		tmp = y * (x_m / z);
	} else if (y <= 2.1e+14) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = y / (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 <= -7.2e+19:
		tmp = y * (x_m / z)
	elif y <= 2.1e+14:
		tmp = (x_m / z) - x_m
	else:
		tmp = y / (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 <= -7.2e+19)
		tmp = Float64(y * Float64(x_m / z));
	elseif (y <= 2.1e+14)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(y / Float64(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 <= -7.2e+19)
		tmp = y * (x_m / z);
	elseif (y <= 2.1e+14)
		tmp = (x_m / z) - x_m;
	else
		tmp = y / (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[LessEqual[y, -7.2e+19], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+14], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -7.2e19

   1. Initial program 83.1%

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

   2. Taylor expanded in y around inf 75.4%

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

      1. associate-/l* 68.9%

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

      2. associate-/r/ 75.5%

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

   4. Applied egg-rr 75.5%

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

   if -7.2e19 < y < 2.1e14

   1. Initial program 90.5%

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

      1. distribute-lft-in 90.5%

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

      2. *-rgt-identity 90.5%

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

   3. Applied egg-rr 90.5%

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

      1. add-cbrt-cube 60.4%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x \cdot \left(y - z\right)\right) \cdot \left(x \cdot \left(y - z\right)\right)\right) \cdot \left(x \cdot \left(y - z\right)\right)}} + x}{z} \]

      2. pow3 60.4%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(x \cdot \left(y - z\right)\right)}^{3}}} + x}{z} \]

   5. Applied egg-rr 60.4%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(x \cdot \left(y - z\right)\right)}^{3}}} + x}{z} \]

   6. Taylor expanded in y around 0 87.2%

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

      1. mul-1-neg 87.2%

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

      2. unsub-neg 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in x around 0 87.2%

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

       1. *-lft-identity 87.2%

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

       2. associate-*l/ 86.9%

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

       3. associate-*r* 82.7%

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

       4. sub-neg 82.7%

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

       5. distribute-rgt-in 74.6%

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

       6. associate-*l/ 74.8%

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

       7. *-lft-identity 74.8%

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

       8. *-lft-identity 74.8%

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

       9. associate-*l/ 74.9%

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

       10. *-lft-identity 74.9%

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

       11. cancel-sign-sub-inv 74.9%

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

       12. *-commutative 74.9%

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

       13. associate-/r/ 96.6%

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

       14. *-inverses 96.6%

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

       15. /-rgt-identity 96.6%

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

   11. Simplified 96.6%

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

   if 2.1e14 < y

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf 77.3%

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

      1. *-commutative 77.3%

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

      2. associate-/l* 81.0%

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

   4. Simplified 81.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 8: 65.1% accurate, 1.3× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 77.1%

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

   2. Taylor expanded in z around inf 82.2%

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

      1. mul-1-neg 82.2%

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

   4. Simplified 82.2%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.5%

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

      1. associate-/l* 94.5%

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

   4. Simplified 94.5%

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

   5. Taylor expanded in y around 0 59.7%

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

4. Final simplification 70.5%

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

Alternative 9: 39.4% 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 88.9%

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

2. Taylor expanded in z around inf 41.3%

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

   1. mul-1-neg 41.3%

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

4. Simplified 41.3%

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

5. Final simplification 41.3%

   \[\leadsto -x \]

Alternative 10: 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 88.9%

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

2. Taylor expanded in z around inf 32.0%

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

   1. associate-*r* 32.0%

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

   2. mul-1-neg 32.0%

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

4. Simplified 32.0%

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

   1. div-inv 31.9%

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

   2. associate-*l* 41.2%

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

   3. div-inv 41.3%

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

   4. *-inverses 41.3%

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

   5. *-commutative 41.3%

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

   6. neg-sub0 41.3%

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

   7. *-un-lft-identity 41.3%

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

   8. sub-neg 41.3%

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

   9. add-sqr-sqrt 20.8%

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

   10. sqrt-unprod 20.9%

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

   11. sqr-neg 20.9%

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

   12. sqrt-unprod 1.7%

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

   13. add-sqr-sqrt 3.1%

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

6. Applied egg-rr 3.1%

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

   1. +-lft-identity 3.1%

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

8. Simplified 3.1%

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

9. Final simplification 3.1%

   \[\leadsto x \]

Developer target: 99.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023334 
(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))