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

Percentage Accurate: 96.1% → 99.8%

Time: 10.4s

Alternatives: 12

Speedup: 0.8×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(z, x\_m \cdot y - x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, x\_m \cdot z, x\_m\right)\\ \end{array} \end{array} \]

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 2.6e-125)
    (fma z (- (* x_m y) x_m) x_m)
    (fma (+ y -1.0) (* x_m z) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.6e-125) {
		tmp = fma(z, ((x_m * y) - x_m), x_m);
	} else {
		tmp = fma((y + -1.0), (x_m * z), x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2.6e-125)
		tmp = fma(z, Float64(Float64(x_m * y) - x_m), x_m);
	else
		tmp = fma(Float64(y + -1.0), Float64(x_m * 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]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2.6e-125], N[(z * N[(N[(x$95$m * y), $MachinePrecision] - x$95$m), $MachinePrecision] + x$95$m), $MachinePrecision], N[(N[(y + -1.0), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision] + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000006e-125

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. distribute-rgt-out-- N/A

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

      12. *-lft-identity N/A

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

      13. unsub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. unsub-neg N/A

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

      18. --lowering--.f64 N/A

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

      19. *-lowering-*.f64 95.2

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

   5. Simplified 95.2%

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

   if 2.60000000000000006e-125 < x

   1. Initial program 97.0%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, x \cdot z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(z \cdot y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -10000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 4 \cdot 10^{+104}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* z y))))
   (*
    x_s
    (if (<= (- 1.0 y) -10000.0)
      t_0
      (if (<= (- 1.0 y) 4e+104) (* 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 * (z * y);
	double tmp;
	if ((1.0 - y) <= -10000.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 4e+104) {
		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 * (z * y)
    if ((1.0d0 - y) <= (-10000.0d0)) then
        tmp = t_0
    else if ((1.0d0 - y) <= 4d+104) 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 * (z * y);
	double tmp;
	if ((1.0 - y) <= -10000.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 4e+104) {
		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 * (z * y)
	tmp = 0
	if (1.0 - y) <= -10000.0:
		tmp = t_0
	elif (1.0 - y) <= 4e+104:
		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(z * y))
	tmp = 0.0
	if (Float64(1.0 - y) <= -10000.0)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 4e+104)
		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 * (z * y);
	tmp = 0.0;
	if ((1.0 - y) <= -10000.0)
		tmp = t_0;
	elseif ((1.0 - y) <= 4e+104)
		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[(z * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(1.0 - y), $MachinePrecision], -10000.0], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 4e+104], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1e4 or 4e104 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 89.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 66.8

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

   5. Simplified 66.8%

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

   if -1e4 < (-.f64 #s(literal 1 binary64) y) < 4e104

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 96.7

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

   5. Simplified 96.7%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, x\_m \cdot y - x\_m, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (fma z (- (* x_m y) x_m) x_m)))
   (*
    x_s
    (if (<= z -2.8e-15) t_0 (if (<= z 2e-19) (fma (* z y) x_m 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 = fma(z, ((x_m * y) - x_m), x_m);
	double tmp;
	if (z <= -2.8e-15) {
		tmp = t_0;
	} else if (z <= 2e-19) {
		tmp = fma((z * y), x_m, 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 = fma(z, Float64(Float64(x_m * y) - x_m), x_m)
	tmp = 0.0
	if (z <= -2.8e-15)
		tmp = t_0;
	elseif (z <= 2e-19)
		tmp = fma(Float64(z * y), x_m, x_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(x$95$m * y), $MachinePrecision] - x$95$m), $MachinePrecision] + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -2.8e-15], t$95$0, If[LessEqual[z, 2e-19], N[(N[(z * y), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.80000000000000014e-15 or 2e-19 < z

   1. Initial program 90.6%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. distribute-rgt-out-- N/A

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

      12. *-lft-identity N/A

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

      13. unsub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. unsub-neg N/A

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

      18. --lowering--.f64 N/A

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

      19. *-lowering-*.f64 99.9

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

   5. Simplified 99.9%

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

   if -2.80000000000000014e-15 < z < 2e-19

   1. Initial program 99.9%

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

   4. Applied egg-rr 94.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. distribute-frac-neg2 N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

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

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

      13. *-commutative N/A

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

      14. flip-- N/A

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

      15. metadata-eval N/A

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

      16. associate-/l* N/A

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

      17. *-rgt-identity N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   9. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - x, x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.7× speedup

Math

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

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 (<= z -1.05)
    (* (+ y -1.0) (* x_m z))
    (if (<= z 1.0) (fma (* z y) x_m x_m) (* z (- (* x_m y) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.05) {
		tmp = (y + -1.0) * (x_m * z);
	} else if (z <= 1.0) {
		tmp = fma((z * y), x_m, x_m);
	} else {
		tmp = z * ((x_m * y) - x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -1.05)
		tmp = Float64(Float64(y + -1.0) * Float64(x_m * z));
	elseif (z <= 1.0)
		tmp = fma(Float64(z * y), x_m, x_m);
	else
		tmp = Float64(z * Float64(Float64(x_m * y) - 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]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -1.05], N[(N[(y + -1.0), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(z * y), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], N[(z * N[(N[(x$95$m * y), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.05000000000000004

   1. Initial program 85.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

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

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

      18. distribute-rgt-out-- N/A

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. --lowering--.f64 N/A

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

      26. *-lowering-*.f64 97.7

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

   5. Simplified 97.7%

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

      1. sub-neg N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 97.9

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

   7. Applied egg-rr 97.9%

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

   if -1.05000000000000004 < z < 1

   1. Initial program 99.9%

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

   4. Applied egg-rr 95.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. distribute-frac-neg2 N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

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

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

      13. *-commutative N/A

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

      14. flip-- N/A

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

      15. metadata-eval N/A

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

      16. associate-/l* N/A

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

      17. *-rgt-identity N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 98.8

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

   9. Simplified 98.8%

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

   if 1 < z

   1. Initial program 93.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

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

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

      18. distribute-rgt-out-- N/A

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. --lowering--.f64 N/A

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

      26. *-lowering-*.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 98.4%

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

Alternative 5: 98.8% accurate, 0.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* z (- (* x_m y) x_m))))
   (* x_s (if (<= z -1.05) t_0 (if (<= z 1.0) (fma (* z y) x_m 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 = z * ((x_m * y) - x_m);
	double tmp;
	if (z <= -1.05) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = fma((z * y), x_m, 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(z * Float64(Float64(x_m * y) - x_m))
	tmp = 0.0
	if (z <= -1.05)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = fma(Float64(z * y), x_m, x_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(x$95$m * y), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.05], t$95$0, If[LessEqual[z, 1.0], N[(N[(z * y), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05000000000000004 or 1 < z

   1. Initial program 90.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

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

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

      18. distribute-rgt-out-- N/A

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. --lowering--.f64 N/A

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

      26. *-lowering-*.f64 98.0

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

   5. Simplified 98.0%

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

   if -1.05000000000000004 < z < 1

   1. Initial program 99.9%

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

   4. Applied egg-rr 95.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. distribute-frac-neg2 N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

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

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

      13. *-commutative N/A

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

      14. flip-- N/A

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

      15. metadata-eval N/A

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

      16. associate-/l* N/A

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

      17. *-rgt-identity N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 98.8

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

   9. Simplified 98.8%

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

4. Final simplification 98.4%

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

Alternative 6: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, x\_m \cdot z, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-15}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (fma y (* x_m z) x_m)))
   (*
    x_s
    (if (<= y -1.35e+19) t_0 (if (<= y 1.8e-15) (* 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 = fma(y, (x_m * z), x_m);
	double tmp;
	if (y <= -1.35e+19) {
		tmp = t_0;
	} else if (y <= 1.8e-15) {
		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 = fma(y, Float64(x_m * z), x_m)
	tmp = 0.0
	if (y <= -1.35e+19)
		tmp = t_0;
	elseif (y <= 1.8e-15)
		tmp = Float64(x_m * Float64(1.0 - z));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(y * N[(x$95$m * z), $MachinePrecision] + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.35e+19], t$95$0, If[LessEqual[y, 1.8e-15], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e19 or 1.8000000000000001e-15 < y

   1. Initial program 90.4%

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

   4. Applied egg-rr 95.6%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 94.8%

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

   if -1.35e19 < y < 1.8000000000000001e-15

   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

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

      1. --lowering--.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot z, x\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \mathsf{fma}\left(y, z, 1\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-15}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (fma y z 1.0))))
   (*
    x_s
    (if (<= y -1.35e+19) t_0 (if (<= y 1.8e-15) (* 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 * fma(y, z, 1.0);
	double tmp;
	if (y <= -1.35e+19) {
		tmp = t_0;
	} else if (y <= 1.8e-15) {
		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 * fma(y, z, 1.0))
	tmp = 0.0
	if (y <= -1.35e+19)
		tmp = t_0;
	elseif (y <= 1.8e-15)
		tmp = Float64(x_m * Float64(1.0 - z));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y * z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.35e+19], t$95$0, If[LessEqual[y, 1.8e-15], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e19 or 1.8000000000000001e-15 < y

   1. Initial program 90.4%

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

   4. Applied egg-rr 95.6%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 94.8%

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

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

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

      4. accelerator-lowering-fma.f64 89.6

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

   9. Applied egg-rr 89.6%

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

   if -1.35e19 < y < 1.8000000000000001e-15

   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

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

      1. --lowering--.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, z, 1\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, z, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \left(x\_m \cdot z\right)\\ \mathbf{elif}\;y \leq 80:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x\_m \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -2.65e+104)
    (* y (* x_m z))
    (if (<= y 80.0) (* x_m (- 1.0 z)) (* z (* x_m y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -2.65e+104) {
		tmp = y * (x_m * z);
	} else if (y <= 80.0) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = z * (x_m * y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.65d+104)) then
        tmp = y * (x_m * z)
    else if (y <= 80.0d0) then
        tmp = x_m * (1.0d0 - z)
    else
        tmp = z * (x_m * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -2.65e+104) {
		tmp = y * (x_m * z);
	} else if (y <= 80.0) {
		tmp = x_m * (1.0 - z);
	} else {
		tmp = z * (x_m * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -2.65e+104:
		tmp = y * (x_m * z)
	elif y <= 80.0:
		tmp = x_m * (1.0 - z)
	else:
		tmp = z * (x_m * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -2.65e+104)
		tmp = Float64(y * Float64(x_m * z));
	elseif (y <= 80.0)
		tmp = Float64(x_m * Float64(1.0 - z));
	else
		tmp = Float64(z * Float64(x_m * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -2.65e+104)
		tmp = y * (x_m * z);
	elseif (y <= 80.0)
		tmp = x_m * (1.0 - z);
	else
		tmp = z * (x_m * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -2.65e+104], N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 80.0], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.6499999999999999e104

   1. Initial program 84.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 65.9

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

   5. Simplified 65.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 78.7

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

   7. Applied egg-rr 78.7%

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

   if -2.6499999999999999e104 < y < 80

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 96.7

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

   5. Simplified 96.7%

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

   if 80 < y

   1. Initial program 92.5%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-lowering-*.f64 72.2

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

   5. Simplified 72.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 80:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := z \cdot \left(x\_m \cdot y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 24:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* z (* x_m y))))
   (* x_s (if (<= y -2.35e+104) t_0 (if (<= y 24.0) (* 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 = z * (x_m * y);
	double tmp;
	if (y <= -2.35e+104) {
		tmp = t_0;
	} else if (y <= 24.0) {
		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 = z * (x_m * y)
    if (y <= (-2.35d+104)) then
        tmp = t_0
    else if (y <= 24.0d0) 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 = z * (x_m * y);
	double tmp;
	if (y <= -2.35e+104) {
		tmp = t_0;
	} else if (y <= 24.0) {
		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 = z * (x_m * y)
	tmp = 0
	if y <= -2.35e+104:
		tmp = t_0
	elif y <= 24.0:
		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(z * Float64(x_m * y))
	tmp = 0.0
	if (y <= -2.35e+104)
		tmp = t_0;
	elseif (y <= 24.0)
		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 = z * (x_m * y);
	tmp = 0.0;
	if (y <= -2.35e+104)
		tmp = t_0;
	elseif (y <= 24.0)
		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[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -2.35e+104], t$95$0, If[LessEqual[y, 24.0], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.35000000000000008e104 or 24 < y

   1. Initial program 89.3%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-lowering-*.f64 73.3

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

   5. Simplified 73.3%

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

   if -2.35000000000000008e104 < y < 24

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 96.7

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

   5. Simplified 96.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+104}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 24:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x_m * z)
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = x_m
    else
        tmp = t_0
    end if
    code = x_s * tmp
end function

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = -(x_m * z);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = x_m;
	else
		tmp = t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 90.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

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

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

      18. distribute-rgt-out-- N/A

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. --lowering--.f64 N/A

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

      26. *-lowering-*.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 54.7

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

   8. Simplified 54.7%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 67.9%

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

4. Final simplification 61.0%

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

Alternative 11: 66.0% accurate, 1.9× speedup

Math

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

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))))

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));
}

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))
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));
}

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))

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 - 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 - 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 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

3. Taylor expanded in y around 0

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

   1. --lowering--.f64 62.4

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

5. Simplified 62.4%

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

Alternative 12: 39.4% accurate, 17.0× 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 94.8%

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

3. Taylor expanded in z around 0

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

5. Simplified 33.9%

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

Developer Target 1: 99.7% accurate, 0.3× 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 2024205 
(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))))