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

Percentage Accurate: 95.9% → 98.0%

Time: 6.3s

Alternatives: 6

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) \cdot x\_m, y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot z\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 2e+27) (fma (* (- z) x_m) y x_m) (* (- 1.0 (* y z)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2e+27) {
		tmp = fma((-z * x_m), y, x_m);
	} else {
		tmp = (1.0 - (y * z)) * x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2e+27)
		tmp = fma(Float64(Float64(-z) * x_m), y, x_m);
	else
		tmp = Float64(Float64(1.0 - Float64(y * z)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2e+27], N[(N[((-z) * x$95$m), $MachinePrecision] * y + x$95$m), $MachinePrecision], N[(N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e27

   1. Initial program 96.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 94.2

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

   4. Applied rewrites 94.2%

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

   if 2e27 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ t_1 := y \cdot \left(\left(-z\right) \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))) (t_1 (* y (* (- z) x_m))))
   (* x_s (if (<= t_0 -0.5) t_1 (if (<= t_0 2.0) (* 1.0 x_m) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double t_1 = y * (-z * x_m);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
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 = 1.0d0 - (y * z)
    t_1 = y * (-z * x_m)
    if (t_0 <= (-0.5d0)) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 * x_m
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double t_1 = y * (-z * x_m);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = 1.0 - (y * z)
	t_1 = y * (-z * x_m)
	tmp = 0
	if t_0 <= -0.5:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = 1.0 * x_m
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	t_1 = Float64(y * Float64(Float64(-z) * x_m))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 * x_m);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = 1.0 - (y * z);
	t_1 = y * (-z * x_m);
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = 1.0 * x_m;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[((-z) * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -0.5], t$95$1, If[LessEqual[t$95$0, 2.0], N[(1.0 * x$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -0.5 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 93.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.3

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 89.6%

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

   if -0.5 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.2%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -0.5:\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;1 - y \cdot z \leq 2:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5:\\ \;\;\;\;\left(\left(-y\right) \cdot z\right) \cdot x\_m\\ \mathbf{elif}\;y \cdot z \leq 0.001:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* y z) -5.0)
    (* (* (- y) z) x_m)
    (if (<= (* y z) 0.001) (* 1.0 x_m) (* y (* (- z) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * z) <= -5.0) {
		tmp = (-y * z) * x_m;
	} else if ((y * z) <= 0.001) {
		tmp = 1.0 * x_m;
	} else {
		tmp = y * (-z * x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
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 * z) <= (-5.0d0)) then
        tmp = (-y * z) * x_m
    else if ((y * z) <= 0.001d0) then
        tmp = 1.0d0 * x_m
    else
        tmp = y * (-z * x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * z) <= -5.0) {
		tmp = (-y * z) * x_m;
	} else if ((y * z) <= 0.001) {
		tmp = 1.0 * x_m;
	} else {
		tmp = y * (-z * x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	tmp = 0
	if (y * z) <= -5.0:
		tmp = (-y * z) * x_m
	elif (y * z) <= 0.001:
		tmp = 1.0 * x_m
	else:
		tmp = y * (-z * x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(y * z) <= -5.0)
		tmp = Float64(Float64(Float64(-y) * z) * x_m);
	elseif (Float64(y * z) <= 0.001)
		tmp = Float64(1.0 * x_m);
	else
		tmp = Float64(y * Float64(Float64(-z) * x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y * z) <= -5.0)
		tmp = (-y * z) * x_m;
	elseif ((y * z) <= 0.001)
		tmp = 1.0 * x_m;
	else
		tmp = y * (-z * x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(y * z), $MachinePrecision], -5.0], N[(N[((-y) * z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 0.001], N[(1.0 * x$95$m), $MachinePrecision], N[(y * N[((-z) * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -5

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if -5 < (*.f64 y z) < 1e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.2%

      \[\leadsto x \cdot \color{blue}{1} \]

   if 1e-3 < (*.f64 y z)

   1. Initial program 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 93.9

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.5

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

   7. Applied rewrites 92.5%

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

   9. Applied rewrites 91.2%

      \[\leadsto -\left(x \cdot z\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5:\\ \;\;\;\;\left(\left(-y\right) \cdot z\right) \cdot x\\ \mathbf{elif}\;y \cdot z \leq 0.001:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5:\\ \;\;\;\;\left(\left(-y\right) \cdot x\_m\right) \cdot z\\ \mathbf{elif}\;y \cdot z \leq 0.001:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* y z) -5.0)
    (* (* (- y) x_m) z)
    (if (<= (* y z) 0.001) (* 1.0 x_m) (* y (* (- z) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * z) <= -5.0) {
		tmp = (-y * x_m) * z;
	} else if ((y * z) <= 0.001) {
		tmp = 1.0 * x_m;
	} else {
		tmp = y * (-z * x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
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 * z) <= (-5.0d0)) then
        tmp = (-y * x_m) * z
    else if ((y * z) <= 0.001d0) then
        tmp = 1.0d0 * x_m
    else
        tmp = y * (-z * x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * z) <= -5.0) {
		tmp = (-y * x_m) * z;
	} else if ((y * z) <= 0.001) {
		tmp = 1.0 * x_m;
	} else {
		tmp = y * (-z * x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	tmp = 0
	if (y * z) <= -5.0:
		tmp = (-y * x_m) * z
	elif (y * z) <= 0.001:
		tmp = 1.0 * x_m
	else:
		tmp = y * (-z * x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(y * z) <= -5.0)
		tmp = Float64(Float64(Float64(-y) * x_m) * z);
	elseif (Float64(y * z) <= 0.001)
		tmp = Float64(1.0 * x_m);
	else
		tmp = Float64(y * Float64(Float64(-z) * x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y * z) <= -5.0)
		tmp = (-y * x_m) * z;
	elseif ((y * z) <= 0.001)
		tmp = 1.0 * x_m;
	else
		tmp = y * (-z * x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(y * z), $MachinePrecision], -5.0], N[(N[((-y) * x$95$m), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 0.001], N[(1.0 * x$95$m), $MachinePrecision], N[(y * N[((-z) * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -5

   1. Initial program 94.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 90.2

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

   4. Applied rewrites 90.2%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if -5 < (*.f64 y z) < 1e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.2%

      \[\leadsto x \cdot \color{blue}{1} \]

   if 1e-3 < (*.f64 y z)

   1. Initial program 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 93.9

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.5

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

   7. Applied rewrites 92.5%

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

   9. Applied rewrites 91.2%

      \[\leadsto -\left(x \cdot z\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5:\\ \;\;\;\;\left(\left(-y\right) \cdot x\right) \cdot z\\ \mathbf{elif}\;y \cdot z \leq 0.001:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := \left(1 - y \cdot z\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+287}:\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (* y z)) x_m)))
   (* x_s (if (<= t_0 -1e+287) (* y (* (- z) x_m)) t_0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (1.0 - (y * z)) * x_m;
	double tmp;
	if (t_0 <= -1e+287) {
		tmp = y * (-z * x_m);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
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 = (1.0d0 - (y * z)) * x_m
    if (t_0 <= (-1d+287)) then
        tmp = y * (-z * 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);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (1.0 - (y * z)) * x_m;
	double tmp;
	if (t_0 <= -1e+287) {
		tmp = y * (-z * x_m);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = (1.0 - (y * z)) * x_m
	tmp = 0
	if t_0 <= -1e+287:
		tmp = y * (-z * x_m)
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(1.0 - Float64(y * z)) * x_m)
	tmp = 0.0
	if (t_0 <= -1e+287)
		tmp = Float64(y * Float64(Float64(-z) * 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);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (1.0 - (y * z)) * x_m;
	tmp = 0.0;
	if (t_0 <= -1e+287)
		tmp = y * (-z * 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -1e+287], N[(y * N[((-z) * x$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < -1.0000000000000001e287

   1. Initial program 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.0

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

   7. Applied rewrites 97.0%

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

   9. Applied rewrites 99.9%

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

   if -1.0000000000000001e287 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z)))

   1. Initial program 98.2%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y \cdot z\right) \cdot x \leq -1 \cdot 10^{+287}:\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.1% accurate, 2.3× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z) :precision binary64 (* x_s (* 1.0 x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (1.0 * x_m);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
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 * (1.0d0 * x_m)
end function

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (1.0 * 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 54.4%

   \[\leadsto x \cdot \color{blue}{1} \]

6. Final simplification 54.4%

   \[\leadsto 1 \cdot x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))