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

Percentage Accurate: 88.5% → 99.8%

Time: 6.0s

Alternatives: 9

Speedup: 1.0×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\frac{x_m \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{y + \left(1 - z\right)}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e-37)
    (/ (* x_m (+ (- y z) 1.0)) z)
    (* x_m (/ (+ y (- 1.0 z)) 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 (x_m <= 2e-37) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = x_m * ((y + (1.0 - z)) / 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 (x_m <= 2d-37) then
        tmp = (x_m * ((y - z) + 1.0d0)) / z
    else
        tmp = x_m * ((y + (1.0d0 - z)) / 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 (x_m <= 2e-37) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = x_m * ((y + (1.0 - z)) / 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 x_m <= 2e-37:
		tmp = (x_m * ((y - z) + 1.0)) / z
	else:
		tmp = x_m * ((y + (1.0 - z)) / 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 (x_m <= 2e-37)
		tmp = Float64(Float64(x_m * Float64(Float64(y - z) + 1.0)) / z);
	else
		tmp = Float64(x_m * Float64(Float64(y + Float64(1.0 - z)) / 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 (x_m <= 2e-37)
		tmp = (x_m * ((y - z) + 1.0)) / z;
	else
		tmp = x_m * ((y + (1.0 - z)) / 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[LessEqual[x$95$m, 2e-37], N[(N[(x$95$m * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(N[(y + N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000013e-37

   1. Initial program 93.3%

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

   if 2.00000000000000013e-37 < x

   1. Initial program 79.1%

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

   3. Taylor expanded in x around 0 79.1%

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

      1. associate--l+ 79.1%

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

      2. +-commutative 79.1%

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

      3. associate-*r/ 99.8%

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

      4. associate-+l- 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + \left(1 - z\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x_m}{z} - x_m\\ t_1 := \frac{x_m \cdot y}{z}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -720000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1580000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{x_m}{\frac{z}{y + 1}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+229}:\\ \;\;\;\;y \cdot \frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- (/ x_m z) x_m)) (t_1 (/ (* x_m y) z)))
   (*
    x_s
    (if (<= y -720000000000.0)
      t_1
      (if (<= y 1580000.0)
        t_0
        (if (<= y 3.2e+84)
          (/ x_m (/ z (+ y 1.0)))
          (if (<= y 5e+139) t_0 (if (<= y 4e+229) (* y (/ x_m z)) t_1))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m / z) - x_m;
	double t_1 = (x_m * y) / z;
	double tmp;
	if (y <= -720000000000.0) {
		tmp = t_1;
	} else if (y <= 1580000.0) {
		tmp = t_0;
	} else if (y <= 3.2e+84) {
		tmp = x_m / (z / (y + 1.0));
	} else if (y <= 5e+139) {
		tmp = t_0;
	} else if (y <= 4e+229) {
		tmp = y * (x_m / z);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x_m / z) - x_m
    t_1 = (x_m * y) / z
    if (y <= (-720000000000.0d0)) then
        tmp = t_1
    else if (y <= 1580000.0d0) then
        tmp = t_0
    else if (y <= 3.2d+84) then
        tmp = x_m / (z / (y + 1.0d0))
    else if (y <= 5d+139) then
        tmp = t_0
    else if (y <= 4d+229) then
        tmp = y * (x_m / z)
    else
        tmp = t_1
    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 = (x_m / z) - x_m;
	double t_1 = (x_m * y) / z;
	double tmp;
	if (y <= -720000000000.0) {
		tmp = t_1;
	} else if (y <= 1580000.0) {
		tmp = t_0;
	} else if (y <= 3.2e+84) {
		tmp = x_m / (z / (y + 1.0));
	} else if (y <= 5e+139) {
		tmp = t_0;
	} else if (y <= 4e+229) {
		tmp = y * (x_m / z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m / z) - x_m
	t_1 = (x_m * y) / z
	tmp = 0
	if y <= -720000000000.0:
		tmp = t_1
	elif y <= 1580000.0:
		tmp = t_0
	elif y <= 3.2e+84:
		tmp = x_m / (z / (y + 1.0))
	elif y <= 5e+139:
		tmp = t_0
	elif y <= 4e+229:
		tmp = y * (x_m / z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m / z) - x_m)
	t_1 = Float64(Float64(x_m * y) / z)
	tmp = 0.0
	if (y <= -720000000000.0)
		tmp = t_1;
	elseif (y <= 1580000.0)
		tmp = t_0;
	elseif (y <= 3.2e+84)
		tmp = Float64(x_m / Float64(z / Float64(y + 1.0)));
	elseif (y <= 5e+139)
		tmp = t_0;
	elseif (y <= 4e+229)
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m / z) - x_m;
	t_1 = (x_m * y) / z;
	tmp = 0.0;
	if (y <= -720000000000.0)
		tmp = t_1;
	elseif (y <= 1580000.0)
		tmp = t_0;
	elseif (y <= 3.2e+84)
		tmp = x_m / (z / (y + 1.0));
	elseif (y <= 5e+139)
		tmp = t_0;
	elseif (y <= 4e+229)
		tmp = y * (x_m / z);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -720000000000.0], t$95$1, If[LessEqual[y, 1580000.0], t$95$0, If[LessEqual[y, 3.2e+84], N[(x$95$m / N[(z / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+139], t$95$0, If[LessEqual[y, 4e+229], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.2e11 or 4e229 < y

   1. Initial program 92.8%

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

   3. Taylor expanded in y around inf 84.4%

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

   if -7.2e11 < y < 1.58e6 or 3.2000000000000001e84 < y < 5.0000000000000003e139

   1. Initial program 87.3%

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

   3. Taylor expanded in x around 0 87.3%

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

      1. associate--l+ 87.3%

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

      2. +-commutative 87.3%

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

      3. associate-*r/ 99.8%

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

      4. associate-+l- 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 85.8%

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

      1. associate-*l/ 87.5%

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

      2. sub-neg 87.5%

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

      3. distribute-rgt-in 81.8%

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

      4. *-lft-identity 81.8%

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

      5. distribute-lft-neg-out 81.8%

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

      6. unsub-neg 81.8%

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

      7. *-commutative 81.8%

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

      8. associate-*l/ 85.9%

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

      9. associate-/l* 98.6%

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

      10. *-inverses 98.6%

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

      11. /-rgt-identity 98.6%

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

   8. Simplified 98.6%

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

   if 1.58e6 < y < 3.2000000000000001e84

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0 73.7%

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

      1. associate-/l* 72.3%

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

   5. Simplified 72.3%

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

   if 5.0000000000000003e139 < y < 4e229

   1. Initial program 84.7%

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

   3. Taylor expanded in y around inf 79.7%

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

      1. associate-/l* 69.8%

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

      2. associate-/r/ 84.7%

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

   5. Simplified 84.7%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -720000000000:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 1580000:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{\frac{z}{y + 1}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+229}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y \cdot \frac{x_m}{z}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-90}:\\ \;\;\;\;\frac{x_m}{z}\\ \mathbf{elif}\;y \leq 15500:\\ \;\;\;\;-x_m\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

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 (/ x_m z))))
   (*
    x_s
    (if (<= y -1.0)
      t_0
      (if (<= y 1.55e-90) (/ x_m z) (if (<= y 15500.0) (- x_m) 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 * (x_m / z);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.55e-90) {
		tmp = x_m / z;
	} else if (y <= 15500.0) {
		tmp = -x_m;
	} else {
		tmp = 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 * (x_m / z)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.55d-90) then
        tmp = x_m / z
    else if (y <= 15500.0d0) then
        tmp = -x_m
    else
        tmp = 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 * (x_m / z);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.55e-90) {
		tmp = x_m / z;
	} else if (y <= 15500.0) {
		tmp = -x_m;
	} else {
		tmp = 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 * (x_m / z)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.55e-90:
		tmp = x_m / z
	elif y <= 15500.0:
		tmp = -x_m
	else:
		tmp = 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(y * Float64(x_m / z))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.55e-90)
		tmp = Float64(x_m / z);
	elseif (y <= 15500.0)
		tmp = Float64(-x_m);
	else
		tmp = 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 * (x_m / z);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.55e-90)
		tmp = x_m / z;
	elseif (y <= 15500.0)
		tmp = -x_m;
	else
		tmp = 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[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.55e-90], N[(x$95$m / z), $MachinePrecision], If[LessEqual[y, 15500.0], (-x$95$m), t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 15500 < y

   1. Initial program 89.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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* 71.8%

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

      2. associate-/r/ 76.0%

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

   5. Simplified 76.0%

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

   if -1 < y < 1.5500000000000001e-90

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0 87.8%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0 63.5%

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

   if 1.5500000000000001e-90 < y < 15500

   1. Initial program 94.3%

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

   3. Taylor expanded in z around inf 67.3%

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

      1. mul-1-neg 67.3%

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

   5. Simplified 67.3%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 15500:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -265000000000 \lor \neg \left(y \leq 3050000\right):\\ \;\;\;\;y \cdot \frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \end{array} \end{array} \]

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 -265000000000.0) (not (<= y 3050000.0)))
    (* 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 <= -265000000000.0) || !(y <= 3050000.0)) {
		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 <= (-265000000000.0d0)) .or. (.not. (y <= 3050000.0d0))) 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 <= -265000000000.0) || !(y <= 3050000.0)) {
		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 <= -265000000000.0) or not (y <= 3050000.0):
		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 <= -265000000000.0) || !(y <= 3050000.0))
		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 <= -265000000000.0) || ~((y <= 3050000.0)))
		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, -265000000000.0], N[Not[LessEqual[y, 3050000.0]], $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 < -2.65e11 or 3.05e6 < y

   1. Initial program 90.2%

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

   3. Taylor expanded in y around inf 78.4%

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

      1. associate-/l* 73.2%

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

      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 -2.65e11 < y < 3.05e6

   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/ 99.8%

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

      4. associate-+l- 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 88.4%

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

      1. associate-*l/ 89.4%

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

      2. sub-neg 89.4%

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

      3. distribute-rgt-in 84.1%

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

      4. *-lft-identity 84.1%

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

      5. distribute-lft-neg-out 84.1%

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

      6. unsub-neg 84.1%

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

      7. *-commutative 84.1%

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

      8. associate-*l/ 88.4%

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

      9. associate-/l* 99.9%

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

      10. *-inverses 99.9%

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

      11. /-rgt-identity 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -265000000000 \lor \neg \left(y \leq 3050000\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -255000000000 \lor \neg \left(y \leq 2750000\right):\\ \;\;\;\;\frac{x_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z} - x_m\\ \end{array} \end{array} \]

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 -255000000000.0) (not (<= y 2750000.0)))
    (/ (* x_m 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 <= -255000000000.0) || !(y <= 2750000.0)) {
		tmp = (x_m * 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 <= (-255000000000.0d0)) .or. (.not. (y <= 2750000.0d0))) then
        tmp = (x_m * 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 <= -255000000000.0) || !(y <= 2750000.0)) {
		tmp = (x_m * 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 <= -255000000000.0) or not (y <= 2750000.0):
		tmp = (x_m * 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 <= -255000000000.0) || !(y <= 2750000.0))
		tmp = Float64(Float64(x_m * 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 <= -255000000000.0) || ~((y <= 2750000.0)))
		tmp = (x_m * 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, -255000000000.0], N[Not[LessEqual[y, 2750000.0]], $MachinePrecision]], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -2.55e11 or 2.75e6 < y

   1. Initial program 90.2%

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

   3. Taylor expanded in y around inf 78.4%

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

   if -2.55e11 < y < 2.75e6

   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/ 99.8%

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

      4. associate-+l- 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 88.4%

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

      1. associate-*l/ 89.4%

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

      2. sub-neg 89.4%

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

      3. distribute-rgt-in 84.1%

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

      4. *-lft-identity 84.1%

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

      5. distribute-lft-neg-out 84.1%

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

      6. unsub-neg 84.1%

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

      7. *-commutative 84.1%

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

      8. associate-*l/ 88.4%

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

      9. associate-/l* 99.9%

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

      10. *-inverses 99.9%

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

      11. /-rgt-identity 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -255000000000 \lor \neg \left(y \leq 2750000\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.2% accurate, 0.7× speedup

Math

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

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 74.0%

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

   3. Taylor expanded in z around inf 72.2%

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

      1. mul-1-neg 72.2%

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

   5. Simplified 72.2%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 57.6%

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

      1. associate-/l* 57.6%

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

   5. Simplified 57.6%

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

   6. Taylor expanded in z around 0 57.2%

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

4. Final simplification 63.3%

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

Alternative 7: 95.9% accurate, 1.0× speedup

Math

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

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)) z))))

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)) / z));
}

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)) / z))
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)) / z));
}

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)) / z))

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in x around 0 89.3%

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

   1. associate--l+ 89.3%

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

   2. +-commutative 89.3%

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

   3. associate-*r/ 95.0%

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

   4. associate-+l- 95.0%

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

5. Simplified 95.0%

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

6. Final simplification 95.0%

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

Alternative 8: 39.3% 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 89.3%

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

3. Taylor expanded in z around inf 31.7%

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

   1. mul-1-neg 31.7%

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

5. Simplified 31.7%

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

6. Final simplification 31.7%

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

Alternative 9: 3.1% 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 89.3%

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

3. Taylor expanded in z around inf 22.9%

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

   1. associate-*r* 22.9%

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

   2. mul-1-neg 22.9%

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

5. Simplified 22.9%

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

   1. div-inv 22.8%

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

   2. associate-*l* 31.6%

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

   3. div-inv 31.7%

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

   4. *-inverses 31.7%

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

   5. *-commutative 31.7%

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

   6. *-un-lft-identity 31.7%

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

   7. neg-sub0 31.7%

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

   8. sub-neg 31.7%

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

   9. add-sqr-sqrt 14.6%

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

   10. sqrt-unprod 16.0%

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

   11. sqr-neg 16.0%

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

   12. sqrt-unprod 1.6%

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

   13. add-sqr-sqrt 3.1%

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

7. Applied egg-rr 3.1%

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

   1. +-lft-identity 3.1%

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

9. Simplified 3.1%

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

10. Final simplification 3.1%

    \[\leadsto x \]
11. Add Preprocessing

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024011 
(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))