Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J

Percentage Accurate: 96.1% → 99.7%

Time: 10.6s

Alternatives: 12

Speedup: 0.6×

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

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * 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 * (1.0d0 - ((1.0d0 - y) * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * 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 * (1.0d0 - ((1.0d0 - y) * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2 \cdot 10^{-33}:\\ \;\;\;\;x\_m + z \cdot \left(x\_m \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m + x\_m \cdot \left(z \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.2e-33)
    (+ x_m (* z (* x_m (+ y -1.0))))
    (+ x_m (* x_m (* z (+ y -1.0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.2e-33) {
		tmp = x_m + (z * (x_m * (y + -1.0)));
	} else {
		tmp = x_m + (x_m * (z * (y + -1.0)));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.2d-33) then
        tmp = x_m + (z * (x_m * (y + (-1.0d0))))
    else
        tmp = x_m + (x_m * (z * (y + (-1.0d0))))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.2e-33) {
		tmp = x_m + (z * (x_m * (y + -1.0)));
	} else {
		tmp = x_m + (x_m * (z * (y + -1.0)));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 1.2e-33:
		tmp = x_m + (z * (x_m * (y + -1.0)))
	else:
		tmp = x_m + (x_m * (z * (y + -1.0)))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.2e-33)
		tmp = Float64(x_m + Float64(z * Float64(x_m * Float64(y + -1.0))));
	else
		tmp = Float64(x_m + Float64(x_m * Float64(z * Float64(y + -1.0))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 1.2e-33)
		tmp = x_m + (z * (x_m * (y + -1.0)));
	else
		tmp = x_m + (x_m * (z * (y + -1.0)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.2e-33], N[(x$95$m + N[(z * N[(x$95$m * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m + N[(x$95$m * N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e-33

   1. Initial program 94.3%

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

   3. Taylor expanded in y around 0 94.3%

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

      1. associate--l+ 94.3%

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

      2. distribute-rgt-in 94.3%

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

      3. *-un-lft-identity 94.3%

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

      4. *-un-lft-identity 94.3%

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

      5. distribute-rgt-out-- 94.3%

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

      6. distribute-rgt1-in 94.3%

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

      7. distribute-lft1-in 94.3%

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

      8. associate-*l* 95.7%

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

      9. sub-neg 95.7%

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

      10. metadata-eval 95.7%

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

   5. Applied egg-rr 95.7%

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

   if 1.2e-33 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-33}:\\ \;\;\;\;x + z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+167} \lor \neg \left(y \leq -3 \cdot 10^{+88} \lor \neg \left(y \leq -1.65 \cdot 10^{+49}\right) \land y \leq 6.8 \cdot 10^{+47}\right):\\ \;\;\;\;x\_m \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

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 (<= y -5.2e+167)
          (not (or (<= y -3e+88) (and (not (<= y -1.65e+49)) (<= y 6.8e+47)))))
    (* x_m (* z y))
    (* x_m (- 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 ((y <= -5.2e+167) || !((y <= -3e+88) || (!(y <= -1.65e+49) && (y <= 6.8e+47)))) {
		tmp = x_m * (z * y);
	} else {
		tmp = x_m * (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 ((y <= (-5.2d+167)) .or. (.not. (y <= (-3d+88)) .or. (.not. (y <= (-1.65d+49))) .and. (y <= 6.8d+47))) then
        tmp = x_m * (z * y)
    else
        tmp = x_m * (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 ((y <= -5.2e+167) || !((y <= -3e+88) || (!(y <= -1.65e+49) && (y <= 6.8e+47)))) {
		tmp = x_m * (z * y);
	} else {
		tmp = x_m * (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 (y <= -5.2e+167) or not ((y <= -3e+88) or (not (y <= -1.65e+49) and (y <= 6.8e+47))):
		tmp = x_m * (z * y)
	else:
		tmp = x_m * (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 ((y <= -5.2e+167) || !((y <= -3e+88) || (!(y <= -1.65e+49) && (y <= 6.8e+47))))
		tmp = Float64(x_m * Float64(z * y));
	else
		tmp = Float64(x_m * Float64(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 ((y <= -5.2e+167) || ~(((y <= -3e+88) || (~((y <= -1.65e+49)) && (y <= 6.8e+47)))))
		tmp = x_m * (z * y);
	else
		tmp = x_m * (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[y, -5.2e+167], N[Not[Or[LessEqual[y, -3e+88], And[N[Not[LessEqual[y, -1.65e+49]], $MachinePrecision], LessEqual[y, 6.8e+47]]]], $MachinePrecision]], N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -5.2000000000000004e167 or -3.00000000000000005e88 < y < -1.6499999999999999e49 or 6.7999999999999996e47 < y

   1. Initial program 89.6%

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

   3. Taylor expanded in y around inf 73.9%

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

      1. *-commutative 73.9%

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

   5. Simplified 73.9%

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

   if -5.2000000000000004e167 < y < -3.00000000000000005e88 or -1.6499999999999999e49 < y < 6.7999999999999996e47

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0 90.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+167} \lor \neg \left(y \leq -3 \cdot 10^{+88} \lor \neg \left(y \leq -1.65 \cdot 10^{+49}\right) \land y \leq 6.8 \cdot 10^{+47}\right):\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+167}:\\ \;\;\;\;x\_m \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{+88} \lor \neg \left(y \leq -4 \cdot 10^{+15}\right) \land y \leq 9.6 \cdot 10^{+48}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x\_m \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -4.5e+167)
    (* x_m (* z y))
    (if (or (<= y -9.8e+88) (and (not (<= y -4e+15)) (<= y 9.6e+48)))
      (* x_m (- 1.0 z))
      (* z (* x_m y))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -4.5e+167) {
		tmp = x_m * (z * y);
	} else if ((y <= -9.8e+88) || (!(y <= -4e+15) && (y <= 9.6e+48))) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = z * (x_m * y);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.5d+167)) then
        tmp = x_m * (z * y)
    else if ((y <= (-9.8d+88)) .or. (.not. (y <= (-4d+15))) .and. (y <= 9.6d+48)) then
        tmp = x_m * (1.0d0 - z)
    else
        tmp = z * (x_m * y)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -4.5e+167) {
		tmp = x_m * (z * y);
	} else if ((y <= -9.8e+88) || (!(y <= -4e+15) && (y <= 9.6e+48))) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = z * (x_m * y);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -4.5e+167:
		tmp = x_m * (z * y)
	elif (y <= -9.8e+88) or (not (y <= -4e+15) and (y <= 9.6e+48)):
		tmp = x_m * (1.0 - z)
	else:
		tmp = z * (x_m * y)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -4.5e+167)
		tmp = Float64(x_m * Float64(z * y));
	elseif ((y <= -9.8e+88) || (!(y <= -4e+15) && (y <= 9.6e+48)))
		tmp = Float64(x_m * Float64(1.0 - z));
	else
		tmp = Float64(z * Float64(x_m * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -4.5e+167)
		tmp = x_m * (z * y);
	elseif ((y <= -9.8e+88) || (~((y <= -4e+15)) && (y <= 9.6e+48)))
		tmp = x_m * (1.0 - z);
	else
		tmp = z * (x_m * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -4.5e+167], N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -9.8e+88], And[N[Not[LessEqual[y, -4e+15]], $MachinePrecision], LessEqual[y, 9.6e+48]]], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.4999999999999999e167

   1. Initial program 93.0%

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

   3. Taylor expanded in y around inf 93.0%

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

      1. *-commutative 93.0%

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

   5. Simplified 93.0%

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

   if -4.4999999999999999e167 < y < -9.8000000000000005e88 or -4e15 < y < 9.6000000000000004e48

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 92.9%

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

   if -9.8000000000000005e88 < y < -4e15 or 9.6000000000000004e48 < y

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf 63.1%

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

      1. associate-*r* 69.2%

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

      2. *-commutative 69.2%

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

   5. Simplified 69.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+167}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{+88} \lor \neg \left(y \leq -4 \cdot 10^{+15}\right) \land y \leq 9.6 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(1 - z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+167}:\\ \;\;\;\;x\_m \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+15}:\\ \;\;\;\;z \cdot \left(x\_m \cdot y\right)\\ \mathbf{elif}\;y \leq 8.7 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x\_m \cdot z\right)\\ \end{array} \end{array} \end{array} \]

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 (- 1.0 z))))
   (*
    x_s
    (if (<= y -4.5e+167)
      (* x_m (* z y))
      (if (<= y -2.6e+88)
        t_0
        (if (<= y -1.65e+15)
          (* z (* x_m y))
          (if (<= y 8.7e+46) t_0 (* y (* 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 t_0 = x_m * (1.0 - z);
	double tmp;
	if (y <= -4.5e+167) {
		tmp = x_m * (z * y);
	} else if (y <= -2.6e+88) {
		tmp = t_0;
	} else if (y <= -1.65e+15) {
		tmp = z * (x_m * y);
	} else if (y <= 8.7e+46) {
		tmp = t_0;
	} else {
		tmp = y * (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) :: t_0
    real(8) :: tmp
    t_0 = x_m * (1.0d0 - z)
    if (y <= (-4.5d+167)) then
        tmp = x_m * (z * y)
    else if (y <= (-2.6d+88)) then
        tmp = t_0
    else if (y <= (-1.65d+15)) then
        tmp = z * (x_m * y)
    else if (y <= 8.7d+46) then
        tmp = t_0
    else
        tmp = y * (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 t_0 = x_m * (1.0 - z);
	double tmp;
	if (y <= -4.5e+167) {
		tmp = x_m * (z * y);
	} else if (y <= -2.6e+88) {
		tmp = t_0;
	} else if (y <= -1.65e+15) {
		tmp = z * (x_m * y);
	} else if (y <= 8.7e+46) {
		tmp = t_0;
	} else {
		tmp = y * (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):
	t_0 = x_m * (1.0 - z)
	tmp = 0
	if y <= -4.5e+167:
		tmp = x_m * (z * y)
	elif y <= -2.6e+88:
		tmp = t_0
	elif y <= -1.65e+15:
		tmp = z * (x_m * y)
	elif y <= 8.7e+46:
		tmp = t_0
	else:
		tmp = y * (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)
	t_0 = Float64(x_m * Float64(1.0 - z))
	tmp = 0.0
	if (y <= -4.5e+167)
		tmp = Float64(x_m * Float64(z * y));
	elseif (y <= -2.6e+88)
		tmp = t_0;
	elseif (y <= -1.65e+15)
		tmp = Float64(z * Float64(x_m * y));
	elseif (y <= 8.7e+46)
		tmp = t_0;
	else
		tmp = Float64(y * 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)
	t_0 = x_m * (1.0 - z);
	tmp = 0.0;
	if (y <= -4.5e+167)
		tmp = x_m * (z * y);
	elseif (y <= -2.6e+88)
		tmp = t_0;
	elseif (y <= -1.65e+15)
		tmp = z * (x_m * y);
	elseif (y <= 8.7e+46)
		tmp = t_0;
	else
		tmp = y * (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_] := Block[{t$95$0 = N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -4.5e+167], N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.6e+88], t$95$0, If[LessEqual[y, -1.65e+15], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.7e+46], t$95$0, N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -4.4999999999999999e167

   1. Initial program 93.0%

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

   3. Taylor expanded in y around inf 93.0%

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

      1. *-commutative 93.0%

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

   5. Simplified 93.0%

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

   if -4.4999999999999999e167 < y < -2.6000000000000001e88 or -1.65e15 < y < 8.69999999999999961e46

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 92.9%

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

   if -2.6000000000000001e88 < y < -1.65e15

   1. Initial program 94.6%

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

   3. Taylor expanded in y around inf 63.1%

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

      1. associate-*r* 68.3%

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

      2. *-commutative 68.3%

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

   5. Simplified 68.3%

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

   if 8.69999999999999961e46 < y

   1. Initial program 86.6%

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

   3. Taylor expanded in y around inf 63.2%

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

      1. *-commutative 63.2%

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

   5. Simplified 63.2%

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

      1. add-cbrt-cube 51.3%

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

      2. pow3 51.2%

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

      3. *-commutative 51.2%

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

      4. associate-*r* 51.2%

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

   7. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(x \cdot y\right) \cdot z\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 69.5%

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

      2. associate-*l* 63.2%

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

      3. *-commutative 63.2%

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

      4. associate-*l* 71.2%

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

   9. Applied egg-rr 71.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+167}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+15}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 8.7 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+167}:\\ \;\;\;\;x\_m \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{+89}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{elif}\;y \leq -9.3 \cdot 10^{+14}:\\ \;\;\;\;z \cdot \left(x\_m \cdot y\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+48}:\\ \;\;\;\;x\_m - x\_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x\_m \cdot z\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -4.8e+167)
    (* x_m (* z y))
    (if (<= y -1.7e+89)
      (* x_m (- 1.0 z))
      (if (<= y -9.3e+14)
        (* z (* x_m y))
        (if (<= y 3.6e+48) (- x_m (* x_m z)) (* y (* 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 (y <= -4.8e+167) {
		tmp = x_m * (z * y);
	} else if (y <= -1.7e+89) {
		tmp = x_m * (1.0 - z);
	} else if (y <= -9.3e+14) {
		tmp = z * (x_m * y);
	} else if (y <= 3.6e+48) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = y * (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 (y <= (-4.8d+167)) then
        tmp = x_m * (z * y)
    else if (y <= (-1.7d+89)) then
        tmp = x_m * (1.0d0 - z)
    else if (y <= (-9.3d+14)) then
        tmp = z * (x_m * y)
    else if (y <= 3.6d+48) then
        tmp = x_m - (x_m * z)
    else
        tmp = y * (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 (y <= -4.8e+167) {
		tmp = x_m * (z * y);
	} else if (y <= -1.7e+89) {
		tmp = x_m * (1.0 - z);
	} else if (y <= -9.3e+14) {
		tmp = z * (x_m * y);
	} else if (y <= 3.6e+48) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = y * (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 y <= -4.8e+167:
		tmp = x_m * (z * y)
	elif y <= -1.7e+89:
		tmp = x_m * (1.0 - z)
	elif y <= -9.3e+14:
		tmp = z * (x_m * y)
	elif y <= 3.6e+48:
		tmp = x_m - (x_m * z)
	else:
		tmp = y * (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 (y <= -4.8e+167)
		tmp = Float64(x_m * Float64(z * y));
	elseif (y <= -1.7e+89)
		tmp = Float64(x_m * Float64(1.0 - z));
	elseif (y <= -9.3e+14)
		tmp = Float64(z * Float64(x_m * y));
	elseif (y <= 3.6e+48)
		tmp = Float64(x_m - Float64(x_m * z));
	else
		tmp = Float64(y * 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 (y <= -4.8e+167)
		tmp = x_m * (z * y);
	elseif (y <= -1.7e+89)
		tmp = x_m * (1.0 - z);
	elseif (y <= -9.3e+14)
		tmp = z * (x_m * y);
	elseif (y <= 3.6e+48)
		tmp = x_m - (x_m * z);
	else
		tmp = y * (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[LessEqual[y, -4.8e+167], N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.7e+89], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -9.3e+14], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e+48], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if y < -4.79999999999999998e167

   1. Initial program 93.0%

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

   3. Taylor expanded in y around inf 93.0%

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

      1. *-commutative 93.0%

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

   5. Simplified 93.0%

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

   if -4.79999999999999998e167 < y < -1.7000000000000001e89

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 79.8%

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

   if -1.7000000000000001e89 < y < -9.3e14

   1. Initial program 94.6%

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

   3. Taylor expanded in y around inf 63.1%

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

      1. associate-*r* 68.3%

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

      2. *-commutative 68.3%

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

   5. Simplified 68.3%

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

   if -9.3e14 < y < 3.59999999999999983e48

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. associate--l+ 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. distribute-rgt1-in 100.0%

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

      7. distribute-lft1-in 100.0%

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

      8. associate-*l* 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 94.2%

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

      1. mul-1-neg 94.2%

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

      2. unsub-neg 94.2%

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

   8. Simplified 94.2%

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

   if 3.59999999999999983e48 < y

   1. Initial program 86.6%

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

   3. Taylor expanded in y around inf 63.2%

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

      1. *-commutative 63.2%

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

   5. Simplified 63.2%

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

      1. add-cbrt-cube 51.3%

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

      2. pow3 51.2%

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

      3. *-commutative 51.2%

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

      4. associate-*r* 51.2%

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

   7. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(x \cdot y\right) \cdot z\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 69.5%

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

      2. associate-*l* 63.2%

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

      3. *-commutative 63.2%

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

      4. associate-*l* 71.2%

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

   9. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot y} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+167}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{elif}\;y \leq -9.3 \cdot 10^{+14}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+48}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(-z\right)\\ t_1 := x\_m \cdot \left(z \cdot y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (- z))) (t_1 (* x_m (* z y))))
   (*
    x_s
    (if (<= z -5e+93)
      t_1
      (if (<= z -8.2e+60)
        t_0
        (if (<= z -1.5e-18) t_1 (if (<= z 1.0) 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 = x_m * -z;
	double t_1 = x_m * (z * y);
	double tmp;
	if (z <= -5e+93) {
		tmp = t_1;
	} else if (z <= -8.2e+60) {
		tmp = t_0;
	} else if (z <= -1.5e-18) {
		tmp = t_1;
	} else if (z <= 1.0) {
		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) :: t_1
    real(8) :: tmp
    t_0 = x_m * -z
    t_1 = x_m * (z * y)
    if (z <= (-5d+93)) then
        tmp = t_1
    else if (z <= (-8.2d+60)) then
        tmp = t_0
    else if (z <= (-1.5d-18)) then
        tmp = t_1
    else if (z <= 1.0d0) 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 = x_m * -z;
	double t_1 = x_m * (z * y);
	double tmp;
	if (z <= -5e+93) {
		tmp = t_1;
	} else if (z <= -8.2e+60) {
		tmp = t_0;
	} else if (z <= -1.5e-18) {
		tmp = t_1;
	} else if (z <= 1.0) {
		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 = x_m * -z
	t_1 = x_m * (z * y)
	tmp = 0
	if z <= -5e+93:
		tmp = t_1
	elif z <= -8.2e+60:
		tmp = t_0
	elif z <= -1.5e-18:
		tmp = t_1
	elif z <= 1.0:
		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(x_m * Float64(-z))
	t_1 = Float64(x_m * Float64(z * y))
	tmp = 0.0
	if (z <= -5e+93)
		tmp = t_1;
	elseif (z <= -8.2e+60)
		tmp = t_0;
	elseif (z <= -1.5e-18)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = 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 = x_m * -z;
	t_1 = x_m * (z * y);
	tmp = 0.0;
	if (z <= -5e+93)
		tmp = t_1;
	elseif (z <= -8.2e+60)
		tmp = t_0;
	elseif (z <= -1.5e-18)
		tmp = t_1;
	elseif (z <= 1.0)
		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[(x$95$m * (-z)), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -5e+93], t$95$1, If[LessEqual[z, -8.2e+60], t$95$0, If[LessEqual[z, -1.5e-18], t$95$1, If[LessEqual[z, 1.0], x$95$m, t$95$0]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.0000000000000001e93 or -8.2e60 < z < -1.49999999999999991e-18

   1. Initial program 89.5%

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

   3. Taylor expanded in y around inf 58.9%

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

      1. *-commutative 58.9%

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

   5. Simplified 58.9%

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

   if -5.0000000000000001e93 < z < -8.2e60 or 1 < z

   1. Initial program 93.5%

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

   3. Taylor expanded in y around 0 93.5%

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

   4. Taylor expanded in z around inf 91.0%

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

      1. sub-neg 91.0%

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

      2. metadata-eval 91.0%

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

      3. associate-*r* 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in y around 0 62.7%

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

      1. mul-1-neg 62.7%

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

      2. distribute-rgt-neg-out 62.7%

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

   9. Simplified 62.7%

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

   if -1.49999999999999991e-18 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 81.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+93}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{+60}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 2 \cdot 10^{+157}:\\ \;\;\;\;x\_m \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot z\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* z (- 1.0 y)) 2e+157)
    (* x_m (+ 1.0 (* z (+ y -1.0))))
    (* (* x_m z) (+ y -1.0)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * (1.0 - y)) <= 2e+157) {
		tmp = x_m * (1.0 + (z * (y + -1.0)));
	} else {
		tmp = (x_m * z) * (y + -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 * (1.0d0 - y)) <= 2d+157) then
        tmp = x_m * (1.0d0 + (z * (y + (-1.0d0))))
    else
        tmp = (x_m * z) * (y + (-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 * (1.0 - y)) <= 2e+157) {
		tmp = x_m * (1.0 + (z * (y + -1.0)));
	} else {
		tmp = (x_m * z) * (y + -1.0);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z * (1.0 - y)) <= 2e+157:
		tmp = x_m * (1.0 + (z * (y + -1.0)))
	else:
		tmp = (x_m * z) * (y + -1.0)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(z * Float64(1.0 - y)) <= 2e+157)
		tmp = Float64(x_m * Float64(1.0 + Float64(z * Float64(y + -1.0))));
	else
		tmp = Float64(Float64(x_m * z) * Float64(y + -1.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z * (1.0 - y)) <= 2e+157)
		tmp = x_m * (1.0 + (z * (y + -1.0)));
	else
		tmp = (x_m * z) * (y + -1.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 2e+157], N[(x$95$m * N[(1.0 + N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * z), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 1 y) z) < 1.99999999999999997e157

   1. Initial program 98.6%

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

   if 1.99999999999999997e157 < (*.f64 (-.f64 1 y) z)

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0 83.6%

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

   4. Taylor expanded in z around inf 83.6%

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

      1. sub-neg 83.6%

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

      2. metadata-eval 83.6%

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

      3. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 2 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-22} \lor \neg \left(z \leq 48000000000000\right):\\ \;\;\;\;x\_m \cdot \left(z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

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 -4.8e-22) (not (<= z 48000000000000.0)))
    (* x_m (* z (+ y -1.0)))
    (* x_m (- 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 <= -4.8e-22) || !(z <= 48000000000000.0)) {
		tmp = x_m * (z * (y + -1.0));
	} else {
		tmp = x_m * (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 <= (-4.8d-22)) .or. (.not. (z <= 48000000000000.0d0))) then
        tmp = x_m * (z * (y + (-1.0d0)))
    else
        tmp = x_m * (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 <= -4.8e-22) || !(z <= 48000000000000.0)) {
		tmp = x_m * (z * (y + -1.0));
	} else {
		tmp = x_m * (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 <= -4.8e-22) or not (z <= 48000000000000.0):
		tmp = x_m * (z * (y + -1.0))
	else:
		tmp = x_m * (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 <= -4.8e-22) || !(z <= 48000000000000.0))
		tmp = Float64(x_m * Float64(z * Float64(y + -1.0)));
	else
		tmp = Float64(x_m * Float64(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 <= -4.8e-22) || ~((z <= 48000000000000.0)))
		tmp = x_m * (z * (y + -1.0));
	else
		tmp = x_m * (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, -4.8e-22], N[Not[LessEqual[z, 48000000000000.0]], $MachinePrecision]], N[(x$95$m * N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4.80000000000000005e-22 or 4.8e13 < z

   1. Initial program 91.1%

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

   3. Taylor expanded in z around inf 88.0%

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

   if -4.80000000000000005e-22 < z < 4.8e13

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 82.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-22} \lor \neg \left(z \leq 48000000000000\right):\\ \;\;\;\;x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-20} \lor \neg \left(z \leq 48000000000000\right):\\ \;\;\;\;\left(x\_m \cdot z\right) \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

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 -8e-20) (not (<= z 48000000000000.0)))
    (* (* x_m z) (+ y -1.0))
    (* x_m (- 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 <= -8e-20) || !(z <= 48000000000000.0)) {
		tmp = (x_m * z) * (y + -1.0);
	} else {
		tmp = x_m * (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 <= (-8d-20)) .or. (.not. (z <= 48000000000000.0d0))) then
        tmp = (x_m * z) * (y + (-1.0d0))
    else
        tmp = x_m * (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 <= -8e-20) || !(z <= 48000000000000.0)) {
		tmp = (x_m * z) * (y + -1.0);
	} else {
		tmp = x_m * (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 <= -8e-20) or not (z <= 48000000000000.0):
		tmp = (x_m * z) * (y + -1.0)
	else:
		tmp = x_m * (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 <= -8e-20) || !(z <= 48000000000000.0))
		tmp = Float64(Float64(x_m * z) * Float64(y + -1.0));
	else
		tmp = Float64(x_m * Float64(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 <= -8e-20) || ~((z <= 48000000000000.0)))
		tmp = (x_m * z) * (y + -1.0);
	else
		tmp = x_m * (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, -8e-20], N[Not[LessEqual[z, 48000000000000.0]], $MachinePrecision]], N[(N[(x$95$m * z), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999956e-20 or 4.8e13 < z

   1. Initial program 91.1%

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

   3. Taylor expanded in y around 0 91.1%

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

   4. Taylor expanded in z around inf 88.0%

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

      1. sub-neg 88.0%

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

      2. metadata-eval 88.0%

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

      3. associate-*r* 96.7%

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

   6. Simplified 96.7%

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

   if -7.99999999999999956e-20 < z < 4.8e13

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 82.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-20} \lor \neg \left(z \leq 48000000000000\right):\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.8% accurate, 0.5× speedup

Math

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

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 -1.0) (not (<= z 1.0)))
    (* (* x_m z) (+ y -1.0))
    (+ x_m (* x_m (* z y))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0 90.8%

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

   4. Taylor expanded in z around inf 88.9%

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

      1. sub-neg 88.9%

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

      2. metadata-eval 88.9%

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

      3. associate-*r* 97.9%

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

   6. Simplified 97.9%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. associate--l+ 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. distribute-rgt-out-- 99.9%

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

      6. distribute-rgt1-in 99.9%

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

      7. distribute-lft1-in 99.9%

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

      8. associate-*l* 89.3%

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

      9. sub-neg 89.3%

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

      10. metadata-eval 89.3%

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

   5. Applied egg-rr 89.3%

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

   6. Taylor expanded in y around inf 99.2%

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

      1. *-commutative 99.2%

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

   8. Simplified 99.2%

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

4. Final simplification 98.6%

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

Alternative 11: 65.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-9} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x\_m \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (<= z -4.8e-9) (not (<= 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 tmp;
	if ((z <= -4.8e-9) || !(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) :: tmp
    if ((z <= (-4.8d-9)) .or. (.not. (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 tmp;
	if ((z <= -4.8e-9) || !(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):
	tmp = 0
	if (z <= -4.8e-9) or not (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)
	tmp = 0.0
	if ((z <= -4.8e-9) || !(z <= 1.0))
		tmp = Float64(x_m * Float64(-z));
	else
		tmp = 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 ((z <= -4.8e-9) || ~((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_] := N[(x$95$s * If[Or[LessEqual[z, -4.8e-9], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x$95$m * (-z)), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4.8e-9 or 1 < z

   1. Initial program 90.9%

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

   3. Taylor expanded in y around 0 90.9%

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

   4. Taylor expanded in z around inf 89.0%

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

      1. sub-neg 89.0%

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

      2. metadata-eval 89.0%

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

      3. associate-*r* 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in y around 0 47.7%

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

      1. mul-1-neg 47.7%

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

      2. distribute-rgt-neg-out 47.7%

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

   9. Simplified 47.7%

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

   if -4.8e-9 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 78.6%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-9} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.9% 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 95.9%

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

3. Taylor expanded in z around 0 44.9%

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

4. Final simplification 44.9%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ t_1 := x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{if}\;t\_0 < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* (- 1.0 y) z))))
        (t_1 (+ x (* (- 1.0 y) (* (- z) x)))))
   (if (< t_0 -1.618195973607049e+50)
     t_1
     (if (< t_0 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = x * (1.0 - ((1.0 - y) * z));
	double t_1 = x + ((1.0 - y) * (-z * x));
	double tmp;
	if (t_0 < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (t_0 < 3.892237649663903e+134) {
		tmp = ((x * y) * z) - ((x * z) - x);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 - ((1.0d0 - y) * z))
    t_1 = x + ((1.0d0 - y) * (-z * x))
    if (t_0 < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (t_0 < 3.892237649663903d+134) then
        tmp = ((x * y) * z) - ((x * z) - x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (1.0 - ((1.0 - y) * z));
	double t_1 = x + ((1.0 - y) * (-z * x));
	double tmp;
	if (t_0 < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (t_0 < 3.892237649663903e+134) {
		tmp = ((x * y) * z) - ((x * z) - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (1.0 - ((1.0 - y) * z))
	t_1 = x + ((1.0 - y) * (-z * x))
	tmp = 0
	if t_0 < -1.618195973607049e+50:
		tmp = t_1
	elif t_0 < 3.892237649663903e+134:
		tmp = ((x * y) * z) - ((x * z) - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
	t_1 = Float64(x + Float64(Float64(1.0 - y) * Float64(Float64(-z) * x)))
	tmp = 0.0
	if (t_0 < -1.618195973607049e+50)
		tmp = t_1;
	elseif (t_0 < 3.892237649663903e+134)
		tmp = Float64(Float64(Float64(x * y) * z) - Float64(Float64(x * z) - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (1.0 - ((1.0 - y) * z));
	t_1 = x + ((1.0 - y) * (-z * x));
	tmp = 0.0;
	if (t_0 < -1.618195973607049e+50)
		tmp = t_1;
	elseif (t_0 < 3.892237649663903e+134)
		tmp = ((x * y) * z) - ((x * z) - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[(1.0 - y), $MachinePrecision] * N[((-z) * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$0, -1.618195973607049e+50], t$95$1, If[Less[t$95$0, 3.892237649663903e+134], N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] - N[(N[(x * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024034 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

  (* x (- 1.0 (* (- 1.0 y) z))))