Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 11.8s

Alternatives: 14

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + 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 - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + 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 - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ y (- x (* (+ y 0.5) (log y)))) z))

C

double code(double x, double y, double z) {
	return (y + (x - ((y + 0.5) * log(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 = (y + (x - ((y + 0.5d0) * log(y)))) - z
end function

Java

public static double code(double x, double y, double z) {
	return (y + (x - ((y + 0.5) * Math.log(y)))) - z;
}

Python

def code(x, y, z):
	return (y + (x - ((y + 0.5) * math.log(y)))) - z

Julia

function code(x, y, z)
	return Float64(Float64(y + Float64(x - Float64(Float64(y + 0.5) * log(y)))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y + (x - ((y + 0.5) * log(y)))) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(y + N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto \left(y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\right) - z \]
4. Add Preprocessing

Alternative 2: 74.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\\ \mathbf{if}\;t\_0 \leq -1.2 \cdot 10^{+201}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+55}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (+ y 0.5) (log y))))))
   (if (<= t_0 -1.2e+201)
     (* y (- 1.0 (log y)))
     (if (<= t_0 -1e+55)
       (- x z)
       (if (<= t_0 500.0) (- (* (log y) -0.5) z) (- x z))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - ((y + 0.5) * log(y)));
	double tmp;
	if (t_0 <= -1.2e+201) {
		tmp = y * (1.0 - log(y));
	} else if (t_0 <= -1e+55) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x - ((y + 0.5d0) * log(y)))
    if (t_0 <= (-1.2d+201)) then
        tmp = y * (1.0d0 - log(y))
    else if (t_0 <= (-1d+55)) then
        tmp = x - z
    else if (t_0 <= 500.0d0) then
        tmp = (log(y) * (-0.5d0)) - z
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y + (x - ((y + 0.5) * Math.log(y)));
	double tmp;
	if (t_0 <= -1.2e+201) {
		tmp = y * (1.0 - Math.log(y));
	} else if (t_0 <= -1e+55) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x - ((y + 0.5) * math.log(y)))
	tmp = 0
	if t_0 <= -1.2e+201:
		tmp = y * (1.0 - math.log(y))
	elif t_0 <= -1e+55:
		tmp = x - z
	elif t_0 <= 500.0:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(Float64(y + 0.5) * log(y))))
	tmp = 0.0
	if (t_0 <= -1.2e+201)
		tmp = Float64(y * Float64(1.0 - log(y)));
	elseif (t_0 <= -1e+55)
		tmp = Float64(x - z);
	elseif (t_0 <= 500.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (x - ((y + 0.5) * log(y)));
	tmp = 0.0;
	if (t_0 <= -1.2e+201)
		tmp = y * (1.0 - log(y));
	elseif (t_0 <= -1e+55)
		tmp = x - z;
	elseif (t_0 <= 500.0)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.2e+201], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e+55], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.19999999999999996e201

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

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

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

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

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

      6. log-lowering-log.f64 73.1

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

   8. Simplified 73.1%

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

   if -1.19999999999999996e201 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.00000000000000001e55 or 500 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 73.6%

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

   if -1.00000000000000001e55 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 500

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 91.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

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

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

      4. log-lowering-log.f64 90.9

         \[\leadsto \color{blue}{\log y} \cdot -0.5 - z \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{\log y \cdot -0.5 - z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -1.2 \cdot 10^{+201}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -1 \cdot 10^{+55}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \log y, x\right)\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (+ y 0.5) (log y))))))
   (if (<= t_0 -2e+96)
     (fma y (- 1.0 (log y)) x)
     (if (<= t_0 20.0)
       (- (fma (log y) (- -0.5 y) y) z)
       (- (fma (log y) -0.5 x) z)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - ((y + 0.5) * log(y)));
	double tmp;
	if (t_0 <= -2e+96) {
		tmp = fma(y, (1.0 - log(y)), x);
	} else if (t_0 <= 20.0) {
		tmp = fma(log(y), (-0.5 - y), y) - z;
	} else {
		tmp = fma(log(y), -0.5, x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(Float64(y + 0.5) * log(y))))
	tmp = 0.0
	if (t_0 <= -2e+96)
		tmp = fma(y, Float64(1.0 - log(y)), x);
	elseif (t_0 <= 20.0)
		tmp = Float64(fma(log(y), Float64(-0.5 - y), y) - z);
	else
		tmp = Float64(fma(log(y), -0.5, x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+96], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 20.0], N[(N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -2.0000000000000001e96

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. log-rec N/A

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

      6. *-lft-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. log-rec N/A

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

      9. sub-neg N/A

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

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

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 88.5

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

   8. Simplified 88.5%

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

   if -2.0000000000000001e96 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 20

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y - \log y \cdot \left(\frac{1}{2} + y\right)\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(y + \left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right) + y\right)} - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right)\right)} + y\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right)} - z \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right) - z \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, y\right) - z \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right), y\right) - z \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2} - y}, y\right) - z \]

      9. --lowering--.f64 94.1

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{-0.5 - y}, y\right) - z \]

   5. Simplified 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)} - z \]

   if 20 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 99.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -2 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \log y, x\right)\\ \mathbf{elif}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq 20:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (+ y 0.5) (log y))))))
   (if (<= t_0 -50.0)
     (- y (fma y (log y) z))
     (if (<= t_0 500.0) (- (* (log y) -0.5) z) (- x z)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - ((y + 0.5) * log(y)));
	double tmp;
	if (t_0 <= -50.0) {
		tmp = y - fma(y, log(y), z);
	} else if (t_0 <= 500.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(Float64(y + 0.5) * log(y))))
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = Float64(y - fma(y, log(y), z));
	elseif (t_0 <= 500.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], N[(y - N[(y * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -50

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. log-lowering-log.f64 74.5

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

   8. Simplified 74.5%

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

   if -50 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 500

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 99.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

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

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

      4. log-lowering-log.f64 99.2

         \[\leadsto \color{blue}{\log y} \cdot -0.5 - z \]

   8. Simplified 99.2%

      \[\leadsto \color{blue}{\log y \cdot -0.5 - z} \]

   if 500 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 98.6%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -50:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \mathbf{elif}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-265}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+133}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e-265)
   (- x z)
   (if (<= y 1.1e-232)
     (fma (log y) -0.5 x)
     (if (<= y 8.5e+133) (- x z) (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e-265) {
		tmp = x - z;
	} else if (y <= 1.1e-232) {
		tmp = fma(log(y), -0.5, x);
	} else if (y <= 8.5e+133) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.2e-265)
		tmp = Float64(x - z);
	elseif (y <= 1.1e-232)
		tmp = fma(log(y), -0.5, x);
	elseif (y <= 8.5e+133)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.2e-265], N[(x - z), $MachinePrecision], If[LessEqual[y, 1.1e-232], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], If[LessEqual[y, 8.5e+133], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.1999999999999996e-265 or 1.10000000000000001e-232 < y < 8.50000000000000044e133

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 73.5%

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

   if 9.1999999999999996e-265 < y < 1.10000000000000001e-232

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{-1}{2} \cdot \log y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} + x \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} \]

      4. log-lowering-log.f64 92.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) \]

   8. Simplified 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} \]

   if 8.50000000000000044e133 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

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

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

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

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

      6. log-lowering-log.f64 83.3

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

   8. Simplified 83.3%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.7e-5) (- (fma (log y) -0.5 x) z) (- (+ y (- x (* y (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.7e-5) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = (y + (x - (y * log(y)))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.7e-5)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(Float64(y + Float64(x - Float64(y * log(y)))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.7e-5], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(y + N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.69999999999999981e-5

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 99.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   if 3.69999999999999981e-5 < y

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -260:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 160:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -260.0) (- x z) (if (<= z 160.0) (fma (log y) -0.5 x) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -260.0) {
		tmp = x - z;
	} else if (z <= 160.0) {
		tmp = fma(log(y), -0.5, x);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -260.0)
		tmp = Float64(x - z);
	elseif (z <= 160.0)
		tmp = fma(log(y), -0.5, x);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -260.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 160.0], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -260 or 160 < z

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 75.9%

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

   if -260 < z < 160

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 61.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 61.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{-1}{2} \cdot \log y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} + x \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} \]

      4. log-lowering-log.f64 60.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) \]

   8. Simplified 60.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\log y, -0.5 - y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e+47)
   (- (fma (log y) -0.5 x) z)
   (+ x (fma (log y) (- -0.5 y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e+47) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = x + fma(log(y), (-0.5 - y), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e+47)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(x + fma(log(y), Float64(-0.5 - y), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e+47], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000008e47

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 96.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   if 8.5000000000000008e47 < y

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{y}{x}\right) - \left(\frac{z}{x} + \frac{\log y \cdot \left(\frac{1}{2} + y\right)}{x}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\frac{y}{x} - \left(\frac{z}{x} + \frac{\log y \cdot \left(\frac{1}{2} + y\right)}{x}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(\frac{y}{x} - \left(\frac{z}{x} + \frac{\log y \cdot \left(\frac{1}{2} + y\right)}{x}\right)\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \color{blue}{x} + x \cdot \left(\frac{y}{x} - \left(\frac{z}{x} + \frac{\log y \cdot \left(\frac{1}{2} + y\right)}{x}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{x} - \left(\frac{z}{x} + \frac{\log y \cdot \left(\frac{1}{2} + y\right)}{x}\right)\right) + x} \]

      5. associate--r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{y}{x} - \frac{z}{x}\right) - \frac{\log y \cdot \left(\frac{1}{2} + y\right)}{x}\right)} + x \]

      6. div-sub N/A

         \[\leadsto x \cdot \left(\color{blue}{\frac{y - z}{x}} - \frac{\log y \cdot \left(\frac{1}{2} + y\right)}{x}\right) + x \]

      7. div-sub N/A

         \[\leadsto x \cdot \color{blue}{\frac{\left(y - z\right) - \log y \cdot \left(\frac{1}{2} + y\right)}{x}} + x \]

      8. associate--r+ N/A

         \[\leadsto x \cdot \frac{\color{blue}{y - \left(z + \log y \cdot \left(\frac{1}{2} + y\right)\right)}}{x} + x \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y - \left(z + \log y \cdot \left(\frac{1}{2} + y\right)\right)}{x}, x\right)} \]

   5. Simplified 59.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y - \mathsf{fma}\left(\log y, y + 0.5, z\right)}{x}, x\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + y\right) - \log y \cdot \left(\frac{1}{2} + y\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(y - \log y \cdot \left(\frac{1}{2} + y\right)\right)} \]

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

         \[\leadsto \color{blue}{x + \left(y - \log y \cdot \left(\frac{1}{2} + y\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(y + \left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right) + y\right)} \]

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

         \[\leadsto x + \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right)\right)} + y\right) \]

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

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right)} \]

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

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right) \]

      8. distribute-neg-in N/A

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, y\right) \]

      9. metadata-eval N/A

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right), y\right) \]

      10. unsub-neg N/A

          \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2} - y}, y\right) \]

      11. --lowering--.f64 86.4

          \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{-0.5 - y}, y\right) \]

   8. Simplified 86.4%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(\log y, -0.5 - y, y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.4 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.4e+47) (- (fma (log y) -0.5 x) z) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.4e+47) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = x + (y * (1.0 - log(y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.4e+47)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.4e+47], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.4e47

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 96.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   if 8.4e47 < y

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. log-rec N/A

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

      6. *-lft-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. log-rec N/A

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

      9. sub-neg N/A

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

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

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 86.4

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

   8. Simplified 86.4%

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

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

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

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

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

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

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

      4. log-lowering-log.f64 86.4

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

   10. Applied egg-rr 86.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.4 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \log y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.2e+48) (- (fma (log y) -0.5 x) z) (fma y (- 1.0 (log y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.2e+48) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = fma(y, (1.0 - log(y)), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.2e+48)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = fma(y, Float64(1.0 - log(y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.2e+48], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2000000000000001e48

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 96.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   if 1.2000000000000001e48 < y

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. log-rec N/A

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

      6. *-lft-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. log-rec N/A

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

      9. sub-neg N/A

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

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

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 86.4

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

   8. Simplified 86.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1 - \log y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.2e+64) (- (fma (log y) -0.5 x) z) (- y (fma y (log y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e+64) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y - fma(y, log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e+64)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y - fma(y, log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.2e+64], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.19999999999999994e64

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 94.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   if 5.19999999999999994e64 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. log-lowering-log.f64 85.7

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

   8. Simplified 85.7%

      \[\leadsto \color{blue}{y - \mathsf{fma}\left(y, \log y, z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 49.2% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+76}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.38e+76) x (if (<= x 2.4e+76) (- 0.0 z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.38e+76) {
		tmp = x;
	} else if (x <= 2.4e+76) {
		tmp = 0.0 - z;
	} else {
		tmp = x;
	}
	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) :: tmp
    if (x <= (-1.38d+76)) then
        tmp = x
    else if (x <= 2.4d+76) then
        tmp = 0.0d0 - z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.38e+76) {
		tmp = x;
	} else if (x <= 2.4e+76) {
		tmp = 0.0 - z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.38e+76:
		tmp = x
	elif x <= 2.4e+76:
		tmp = 0.0 - z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.38e+76)
		tmp = x;
	elseif (x <= 2.4e+76)
		tmp = Float64(0.0 - z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.38e+76)
		tmp = x;
	elseif (x <= 2.4e+76)
		tmp = 0.0 - z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.38e+76], x, If[LessEqual[x, 2.4e+76], N[(0.0 - z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3800000000000001e76 or 2.4e76 < x

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 68.2%

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

   if -1.3800000000000001e76 < x < 2.4e76

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 37.0

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

   5. Simplified 37.0%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 37.0

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

   7. Applied egg-rr 37.0%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+76}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.5% accurate, 29.5× speedup

Math

\[\begin{array}{l} \\ x - z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Simplified 54.0%

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

Alternative 14: 30.9% accurate, 118.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

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

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
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Simplified 28.1%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- (+ y x) z) (* (+ y 0.5) (log y))))

C

double code(double x, double y, double z) {
	return ((y + x) - z) - ((y + 0.5) * log(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((y + x) - z) - ((y + 0.5d0) * log(y))
end function

Java

public static double code(double x, double y, double z) {
	return ((y + x) - z) - ((y + 0.5) * Math.log(y));
}

Python

def code(x, y, z):
	return ((y + x) - z) - ((y + 0.5) * math.log(y))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y + x) - z) - Float64(Float64(y + 0.5) * log(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y + x) - z) - ((y + 0.5) * log(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y + x), $MachinePrecision] - z), $MachinePrecision] - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024196 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- (+ y x) z) (* (+ y 1/2) (log y))))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))