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

Percentage Accurate: 96.2% → 99.8%

Time: 9.8s

Alternatives: 10

Speedup: 1.0×

• 21.1% 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: 96.2% 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: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-51}:\\ \;\;\;\;x\_m + z \cdot \left(x\_m \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 + \left(y \cdot z - z\right)\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-51)
    (+ x_m (* z (* x_m (+ y -1.0))))
    (* x_m (+ 1.0 (- (* y 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 <= 5e-51) {
		tmp = x_m + (z * (x_m * (y + -1.0)));
	} else {
		tmp = x_m * (1.0 + ((y * 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 <= 5d-51) then
        tmp = x_m + (z * (x_m * (y + (-1.0d0))))
    else
        tmp = x_m * (1.0d0 + ((y * 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 <= 5e-51) {
		tmp = x_m + (z * (x_m * (y + -1.0)));
	} else {
		tmp = x_m * (1.0 + ((y * 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 <= 5e-51:
		tmp = x_m + (z * (x_m * (y + -1.0)))
	else:
		tmp = x_m * (1.0 + ((y * 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 <= 5e-51)
		tmp = Float64(x_m + Float64(z * Float64(x_m * Float64(y + -1.0))));
	else
		tmp = Float64(x_m * Float64(1.0 + Float64(Float64(y * 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 <= 5e-51)
		tmp = x_m + (z * (x_m * (y + -1.0)));
	else
		tmp = x_m * (1.0 + ((y * 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, 5e-51], N[(x$95$m + N[(z * N[(x$95$m * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 + N[(N[(y * z), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000004e-51

   1. Initial program 97.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. fma-define N/A

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

      4. distribute-lft-neg-out N/A

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

      5. fmm-undef N/A

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

      6. *-lft-identity N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(\left(1 - y\right) \cdot z\right) \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\left(\left(1 - y\right) \cdot z\right), \color{blue}{x}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(1 - y\right), z\right), x\right)\right) \]

      10. --lowering--.f64 97.0%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), z\right), x\right)\right) \]

   4. Applied egg-rr 97.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(x \cdot \color{blue}{\left(\left(1 - y\right) \cdot z\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\left(x \cdot \left(1 - y\right)\right) \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot \left(1 - y\right)\right), \color{blue}{z}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 - y\right)\right), z\right)\right) \]

      5. --lowering--.f64 96.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, y\right)\right), z\right)\right) \]

   6. Applied egg-rr 96.5%

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

   if 5.00000000000000004e-51 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot \color{blue}{\left(1 - y\right)}\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot 1 + \color{blue}{z \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{fma}\left(z, \color{blue}{1}, z \cdot \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{fma}\left(z, 1, \mathsf{neg}\left(z \cdot y\right)\right)\right)\right)\right) \]

      6. fmm-undef N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot 1 - \color{blue}{z \cdot y}\right)\right)\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z - \color{blue}{z} \cdot y\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(z, \color{blue}{\left(z \cdot y\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-51}:\\ \;\;\;\;x + z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot z - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(0 - z\right)\\ t_1 := x\_m \cdot \left(y \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -0.0105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-98}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (- 0.0 z))) (t_1 (* x_m (* y z))))
   (*
    x_s
    (if (<= z -1.25e+95)
      t_1
      (if (<= z -0.0105)
        t_0
        (if (<= z 4.5e-98) x_m (if (<= z 6.6e+100) t_1 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 * (0.0 - z);
	double t_1 = x_m * (y * z);
	double tmp;
	if (z <= -1.25e+95) {
		tmp = t_1;
	} else if (z <= -0.0105) {
		tmp = t_0;
	} else if (z <= 4.5e-98) {
		tmp = x_m;
	} else if (z <= 6.6e+100) {
		tmp = t_1;
	} 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 * (0.0d0 - z)
    t_1 = x_m * (y * z)
    if (z <= (-1.25d+95)) then
        tmp = t_1
    else if (z <= (-0.0105d0)) then
        tmp = t_0
    else if (z <= 4.5d-98) then
        tmp = x_m
    else if (z <= 6.6d+100) then
        tmp = t_1
    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 * (0.0 - z);
	double t_1 = x_m * (y * z);
	double tmp;
	if (z <= -1.25e+95) {
		tmp = t_1;
	} else if (z <= -0.0105) {
		tmp = t_0;
	} else if (z <= 4.5e-98) {
		tmp = x_m;
	} else if (z <= 6.6e+100) {
		tmp = t_1;
	} 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 * (0.0 - z)
	t_1 = x_m * (y * z)
	tmp = 0
	if z <= -1.25e+95:
		tmp = t_1
	elif z <= -0.0105:
		tmp = t_0
	elif z <= 4.5e-98:
		tmp = x_m
	elif z <= 6.6e+100:
		tmp = t_1
	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(0.0 - z))
	t_1 = Float64(x_m * Float64(y * z))
	tmp = 0.0
	if (z <= -1.25e+95)
		tmp = t_1;
	elseif (z <= -0.0105)
		tmp = t_0;
	elseif (z <= 4.5e-98)
		tmp = x_m;
	elseif (z <= 6.6e+100)
		tmp = t_1;
	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 * (0.0 - z);
	t_1 = x_m * (y * z);
	tmp = 0.0;
	if (z <= -1.25e+95)
		tmp = t_1;
	elseif (z <= -0.0105)
		tmp = t_0;
	elseif (z <= 4.5e-98)
		tmp = x_m;
	elseif (z <= 6.6e+100)
		tmp = t_1;
	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 * N[(0.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.25e+95], t$95$1, If[LessEqual[z, -0.0105], t$95$0, If[LessEqual[z, 4.5e-98], x$95$m, If[LessEqual[z, 6.6e+100], t$95$1, t$95$0]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25000000000000006e95 or 4.49999999999999997e-98 < z < 6.6000000000000002e100

   1. Initial program 98.5%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 67.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 67.0%

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

   if -1.25000000000000006e95 < z < -0.0105000000000000007 or 6.6000000000000002e100 < z

   1. Initial program 92.7%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1 + \left(1 - y\right) \cdot z}{1 \cdot 1 - \left(\left(1 - y\right) \cdot z\right) \cdot \left(\left(1 - y\right) \cdot z\right)}\right)}\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 - \left(\left(1 - y\right) \cdot z\right) \cdot \left(\left(1 - y\right) \cdot z\right)}{1 + \left(1 - y\right) \cdot z}}}\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{1 - \color{blue}{\left(1 - y\right) \cdot z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 - \left(1 - y\right) \cdot z\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\left(1 - y\right) \cdot z\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(1 - y\right), \color{blue}{z}\right)\right)\right)\right) \]

      10. --lowering--.f64 92.5%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), z\right)\right)\right)\right) \]

   4. Applied egg-rr 92.5%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{-1}{z \cdot \left(1 - y\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{-1}{\left(1 - y\right) \cdot \color{blue}{z}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\frac{-1}{1 - y}}{\color{blue}{z}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{-1}{1 - y}\right), \color{blue}{z}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \left(1 - y\right)\right), z\right)\right) \]

      5. --lowering--.f64 91.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(1, y\right)\right), z\right)\right) \]

   7. Simplified 91.5%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{-1}{z}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 67.3%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{z}\right)\right) \]

   10. Simplified 67.3%

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

       1. div-inv N/A

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

       2. frac-2neg N/A

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

       3. metadata-eval N/A

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

       4. remove-double-div N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(z\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{neg}\left(x \cdot z\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(x \cdot z\right)\right) \]

       7. *-lowering-*.f64 67.4%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, z\right)\right) \]

   12. Applied egg-rr 67.4%

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

   if -0.0105000000000000007 < z < 4.49999999999999997e-98

   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

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

   5. Simplified 80.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+95}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -0.0105:\\ \;\;\;\;x \cdot \left(0 - z\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-98}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+100}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot z\right) \cdot \left(y + -1\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.28:\\ \;\;\;\;x\_m + x\_m \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (* x_m z) (+ y -1.0))))
   (* x_s (if (<= z -1.05) t_0 (if (<= z 0.28) (+ x_m (* x_m (* y z))) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * z) * (y + -1.0);
	double tmp;
	if (z <= -1.05) {
		tmp = t_0;
	} else if (z <= 0.28) {
		tmp = x_m + (x_m * (y * z));
	} 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 = (x_m * z) * (y + (-1.0d0))
    if (z <= (-1.05d0)) then
        tmp = t_0
    else if (z <= 0.28d0) then
        tmp = x_m + (x_m * (y * z))
    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) * (y + -1.0);
	double tmp;
	if (z <= -1.05) {
		tmp = t_0;
	} else if (z <= 0.28) {
		tmp = x_m + (x_m * (y * z));
	} 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) * (y + -1.0)
	tmp = 0
	if z <= -1.05:
		tmp = t_0
	elif z <= 0.28:
		tmp = x_m + (x_m * (y * z))
	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(Float64(x_m * z) * Float64(y + -1.0))
	tmp = 0.0
	if (z <= -1.05)
		tmp = t_0;
	elseif (z <= 0.28)
		tmp = Float64(x_m + Float64(x_m * Float64(y * z)));
	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) * (y + -1.0);
	tmp = 0.0;
	if (z <= -1.05)
		tmp = t_0;
	elseif (z <= 0.28)
		tmp = x_m + (x_m * (y * z));
	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[(N[(x$95$m * z), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.05], t$95$0, If[LessEqual[z, 0.28], N[(x$95$m + N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05000000000000004 or 0.28000000000000003 < z

   1. Initial program 95.1%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1 + \left(1 - y\right) \cdot z}{1 \cdot 1 - \left(\left(1 - y\right) \cdot z\right) \cdot \left(\left(1 - y\right) \cdot z\right)}\right)}\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 - \left(\left(1 - y\right) \cdot z\right) \cdot \left(\left(1 - y\right) \cdot z\right)}{1 + \left(1 - y\right) \cdot z}}}\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{1 - \color{blue}{\left(1 - y\right) \cdot z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 - \left(1 - y\right) \cdot z\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\left(1 - y\right) \cdot z\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(1 - y\right), \color{blue}{z}\right)\right)\right)\right) \]

      10. --lowering--.f64 95.0%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), z\right)\right)\right)\right) \]

   4. Applied egg-rr 95.0%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{-1}{z \cdot \left(1 - y\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{-1}{\left(1 - y\right) \cdot \color{blue}{z}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\frac{-1}{1 - y}}{\color{blue}{z}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{-1}{1 - y}\right), \color{blue}{z}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \left(1 - y\right)\right), z\right)\right) \]

      5. --lowering--.f64 94.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(1, y\right)\right), z\right)\right) \]

   7. Simplified 94.5%

      \[\leadsto \frac{x}{\color{blue}{\frac{\frac{-1}{1 - y}}{z}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/l/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. remove-double-div N/A

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

      9. distribute-lft-neg-in N/A

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

      10. neg-sub0 N/A

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

      11. associate--r- N/A

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

      12. metadata-eval N/A

          \[\leadsto \left(-1 + y\right) \cdot \left(x \cdot z\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(-1 + y\right), \color{blue}{\left(x \cdot z\right)}\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y + -1\right), \left(\color{blue}{x} \cdot z\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, -1\right), \left(\color{blue}{x} \cdot z\right)\right) \]

      16. *-lowering-*.f64 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, -1\right), \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right) \]

   9. Applied egg-rr 99.4%

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

   if -1.05000000000000004 < z < 0.28000000000000003

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. fma-define N/A

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

      4. distribute-lft-neg-out N/A

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

      5. fmm-undef N/A

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

      6. *-lft-identity N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(\left(1 - y\right) \cdot z\right) \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\left(\left(1 - y\right) \cdot z\right), \color{blue}{x}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(1 - y\right), z\right), x\right)\right) \]

      10. --lowering--.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), z\right), x\right)\right) \]

   4. Applied egg-rr 99.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\left(1 - y\right) \cdot \color{blue}{\left(z \cdot x\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\left(1 - y\right), \color{blue}{\left(z \cdot x\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), \left(x \cdot \color{blue}{z}\right)\right)\right) \]

      5. *-lowering-*.f64 97.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   6. Applied egg-rr 97.5%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(x \cdot \left(y \cdot z\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(0 - \color{blue}{x \cdot \left(y \cdot z\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{\left(x \cdot \left(y \cdot z\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 98.8%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right)\right)\right) \]

   9. Simplified 98.8%

      \[\leadsto x - \color{blue}{\left(0 - x \cdot \left(z \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. associate--r- N/A

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

       2. --rgt-identity N/A

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

       3. +-commutative N/A

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

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(z \cdot y\right)\right), \color{blue}{x}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(z \cdot y\right)\right), x\right) \]

       6. *-lowering-*.f64 98.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, y\right)\right), x\right) \]

   11. Applied egg-rr 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;z \leq 0.28:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+16}:\\ \;\;\;\;x\_m \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+39}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x\_m \cdot z\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -3e+16)
    (* x_m (* y z))
    (if (<= y 1.06e+39) (* x_m (- 1.0 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 <= -3e+16) {
		tmp = x_m * (y * z);
	} else if (y <= 1.06e+39) {
		tmp = x_m * (1.0 - 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 <= (-3d+16)) then
        tmp = x_m * (y * z)
    else if (y <= 1.06d+39) then
        tmp = x_m * (1.0d0 - 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 <= -3e+16) {
		tmp = x_m * (y * z);
	} else if (y <= 1.06e+39) {
		tmp = x_m * (1.0 - 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 <= -3e+16:
		tmp = x_m * (y * z)
	elif y <= 1.06e+39:
		tmp = x_m * (1.0 - 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 <= -3e+16)
		tmp = Float64(x_m * Float64(y * z));
	elseif (y <= 1.06e+39)
		tmp = Float64(x_m * Float64(1.0 - 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 <= -3e+16)
		tmp = x_m * (y * z);
	elseif (y <= 1.06e+39)
		tmp = x_m * (1.0 - 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, -3e+16], N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e+39], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3e16

   1. Initial program 98.2%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 84.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 84.1%

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

   if -3e16 < y < 1.06000000000000005e39

   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

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 - z\right)}\right) \]

      2. --lowering--.f64 95.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{z}\right)\right) \]

   5. Simplified 95.9%

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

   if 1.06000000000000005e39 < y

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 68.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 68.2%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot z\right), \color{blue}{y}\right) \]

      3. *-lowering-*.f64 77.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 77.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(y \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+37}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (* y z))))
   (* x_s (if (<= y -1.1e+17) t_0 (if (<= y 3e+37) (* x_m (- 1.0 z)) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = x_m * (y * z);
	double tmp;
	if (y <= -1.1e+17) {
		tmp = t_0;
	} else if (y <= 3e+37) {
		tmp = x_m * (1.0 - z);
	} 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 = x_m * (y * z)
    if (y <= (-1.1d+17)) then
        tmp = t_0
    else if (y <= 3d+37) then
        tmp = x_m * (1.0d0 - z)
    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 * (y * z);
	double tmp;
	if (y <= -1.1e+17) {
		tmp = t_0;
	} else if (y <= 3e+37) {
		tmp = x_m * (1.0 - z);
	} 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 * (y * z)
	tmp = 0
	if y <= -1.1e+17:
		tmp = t_0
	elif y <= 3e+37:
		tmp = x_m * (1.0 - z)
	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(y * z))
	tmp = 0.0
	if (y <= -1.1e+17)
		tmp = t_0;
	elseif (y <= 3e+37)
		tmp = Float64(x_m * Float64(1.0 - z));
	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 * (y * z);
	tmp = 0.0;
	if (y <= -1.1e+17)
		tmp = t_0;
	elseif (y <= 3e+37)
		tmp = x_m * (1.0 - z);
	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 * N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.1e+17], t$95$0, If[LessEqual[y, 3e+37], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e17 or 3.00000000000000022e37 < y

   1. Initial program 94.7%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   5. Simplified 77.2%

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

   if -1.1e17 < y < 3.00000000000000022e37

   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

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 - z\right)}\right) \]

      2. --lowering--.f64 95.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{z}\right)\right) \]

   5. Simplified 95.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(0 - z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.0105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+19}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (- 0.0 z))))
   (* x_s (if (<= z -0.0105) t_0 (if (<= z 6e+19) 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 * (0.0 - z);
	double tmp;
	if (z <= -0.0105) {
		tmp = t_0;
	} else if (z <= 6e+19) {
		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 = x_m * (0.0d0 - z)
    if (z <= (-0.0105d0)) then
        tmp = t_0
    else if (z <= 6d+19) 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 * (0.0 - z);
	double tmp;
	if (z <= -0.0105) {
		tmp = t_0;
	} else if (z <= 6e+19) {
		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 * (0.0 - z)
	tmp = 0
	if z <= -0.0105:
		tmp = t_0
	elif z <= 6e+19:
		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(0.0 - z))
	tmp = 0.0
	if (z <= -0.0105)
		tmp = t_0;
	elseif (z <= 6e+19)
		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 * (0.0 - z);
	tmp = 0.0;
	if (z <= -0.0105)
		tmp = t_0;
	elseif (z <= 6e+19)
		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 * N[(0.0 - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -0.0105], t$95$0, If[LessEqual[z, 6e+19], x$95$m, t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0105000000000000007 or 6e19 < z

   1. Initial program 95.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1 + \left(1 - y\right) \cdot z}{1 \cdot 1 - \left(\left(1 - y\right) \cdot z\right) \cdot \left(\left(1 - y\right) \cdot z\right)}\right)}\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 - \left(\left(1 - y\right) \cdot z\right) \cdot \left(\left(1 - y\right) \cdot z\right)}{1 + \left(1 - y\right) \cdot z}}}\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{1 - \color{blue}{\left(1 - y\right) \cdot z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 - \left(1 - y\right) \cdot z\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\left(1 - y\right) \cdot z\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(1 - y\right), \color{blue}{z}\right)\right)\right)\right) \]

      10. --lowering--.f64 94.9%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), z\right)\right)\right)\right) \]

   4. Applied egg-rr 94.9%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{-1}{z \cdot \left(1 - y\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{-1}{\left(1 - y\right) \cdot \color{blue}{z}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\frac{-1}{1 - y}}{\color{blue}{z}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{-1}{1 - y}\right), \color{blue}{z}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \left(1 - y\right)\right), z\right)\right) \]

      5. --lowering--.f64 94.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(1, y\right)\right), z\right)\right) \]

   7. Simplified 94.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{-1}{z}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 58.3%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{z}\right)\right) \]

   10. Simplified 58.3%

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

       1. div-inv N/A

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

       2. frac-2neg N/A

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

       3. metadata-eval N/A

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

       4. remove-double-div N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(z\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{neg}\left(x \cdot z\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(x \cdot z\right)\right) \]

       7. *-lowering-*.f64 58.4%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, z\right)\right) \]

   12. Applied egg-rr 58.4%

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

   if -0.0105000000000000007 < z < 6e19

   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

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

   5. Simplified 73.9%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0105:\\ \;\;\;\;x \cdot \left(0 - z\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (+ 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) {
	return x_s * (x_m + ((x_m * z) * (y + -1.0)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. fma-define N/A

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

   4. distribute-lft-neg-out N/A

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

   5. fmm-undef N/A

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

   6. *-lft-identity N/A

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

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(\left(1 - y\right) \cdot z\right) \cdot x\right)}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\left(\left(1 - y\right) \cdot z\right), \color{blue}{x}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(1 - y\right), z\right), x\right)\right) \]

   10. --lowering--.f64 97.7%

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), z\right), x\right)\right) \]

4. Applied egg-rr 97.7%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(\left(1 - y\right) \cdot \color{blue}{\left(z \cdot x\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\left(1 - y\right), \color{blue}{\left(z \cdot x\right)}\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), \left(\color{blue}{z} \cdot x\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), \left(x \cdot \color{blue}{z}\right)\right)\right) \]

   5. *-lowering-*.f64 98.6%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 8: 96.2% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (* x_m (+ 1.0 (- (* y 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 * (1.0 + ((y * 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 * (1.0d0 + ((y * 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 * (1.0 + ((y * 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 * (1.0 + ((y * 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(1.0 + Float64(Float64(y * 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 * (1.0 + ((y * 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[(1.0 + N[(N[(y * z), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.7%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot \color{blue}{\left(1 - y\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot 1 + \color{blue}{z \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{fma}\left(z, \color{blue}{1}, z \cdot \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

   5. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{fma}\left(z, 1, \mathsf{neg}\left(z \cdot y\right)\right)\right)\right)\right) \]

   6. fmm-undef N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z \cdot 1 - \color{blue}{z \cdot y}\right)\right)\right) \]

   7. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(z - \color{blue}{z} \cdot y\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(z, \color{blue}{\left(z \cdot y\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 97.7%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right)\right)\right) \]

4. Applied egg-rr 97.7%

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

5. Final simplification 97.7%

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

Alternative 9: 96.2% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (* x_m (+ 1.0 (* 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) {
	return x_s * (x_m * (1.0 + (z * (y + -1.0))));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Final simplification 97.7%

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

Alternative 10: 38.1% 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 #s(literal 1 binary64) 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 97.7%

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

3. Taylor expanded in z around 0

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

5. Simplified 42.1%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 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 2024140 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- 1 (* (- 1 y) z))) -161819597360704900000000000000000000000000000000000) (+ x (* (- 1 y) (* (- z) x))) (if (< (* x (- 1 (* (- 1 y) z))) 389223764966390300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1 y) (* (- z) x))))))

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