System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2

Percentage Accurate: 99.9% → 99.9%

Time: 11.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(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 * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + Math.log(z)));
}

Python

def code(x, y, z):
	return (x * 0.5) + (y * ((1.0 - z) + math.log(z)))

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(y * Float64(Float64(1.0 - z) + log(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.5) + (y * ((1.0 - z) + log(z)));
end

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(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 * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + Math.log(z)));
}

Python

def code(x, y, z):
	return (x * 0.5) + (y * ((1.0 - z) + math.log(z)))

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(y * Float64(Float64(1.0 - z) + log(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.5) + (y * ((1.0 - z) + log(z)));
end

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \left(1 - z\right) + \log z, x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma y (+ (- 1.0 z) (log z)) (* x 0.5)))

C

double code(double x, double y, double z) {
	return fma(y, ((1.0 - z) + log(z)), (x * 0.5));
}

Julia

function code(x, y, z)
	return fma(y, Float64(Float64(1.0 - z) + log(z)), Float64(x * 0.5))
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Step-by-step derivation

   1. +-commutative 99.9%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right) + x \cdot 0.5} \]

   2. fma-define 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(1 - z\right) + \log z, x \cdot 0.5\right)} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(1 - z\right) + \log z, x \cdot 0.5\right)} \]
4. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(y, \left(1 - z\right) + \log z, x \cdot 0.5\right) \]
6. Add Preprocessing

Alternative 2: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + y \cdot \log z\\ t_1 := x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{if}\;z \leq 5.5 \cdot 10^{-297}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-264}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-256}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-190}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (* y (log z)))) (t_1 (* x (- 0.5 (* z (/ y x))))))
   (if (<= z 5.5e-297)
     t_1
     (if (<= z 1e-264)
       t_0
       (if (<= z 2.25e-256)
         (* x 0.5)
         (if (<= z 7e-190)
           (* y (+ 1.0 (log z)))
           (if (<= z 6.2e-157)
             t_1
             (if (<= z 7.2e-6) t_0 (fma y (- z) (* x 0.5))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (y * log(z));
	double t_1 = x * (0.5 - (z * (y / x)));
	double tmp;
	if (z <= 5.5e-297) {
		tmp = t_1;
	} else if (z <= 1e-264) {
		tmp = t_0;
	} else if (z <= 2.25e-256) {
		tmp = x * 0.5;
	} else if (z <= 7e-190) {
		tmp = y * (1.0 + log(z));
	} else if (z <= 6.2e-157) {
		tmp = t_1;
	} else if (z <= 7.2e-6) {
		tmp = t_0;
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(y * log(z)))
	t_1 = Float64(x * Float64(0.5 - Float64(z * Float64(y / x))))
	tmp = 0.0
	if (z <= 5.5e-297)
		tmp = t_1;
	elseif (z <= 1e-264)
		tmp = t_0;
	elseif (z <= 2.25e-256)
		tmp = Float64(x * 0.5);
	elseif (z <= 7e-190)
		tmp = Float64(y * Float64(1.0 + log(z)));
	elseif (z <= 6.2e-157)
		tmp = t_1;
	elseif (z <= 7.2e-6)
		tmp = t_0;
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(0.5 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.5e-297], t$95$1, If[LessEqual[z, 1e-264], t$95$0, If[LessEqual[z, 2.25e-256], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 7e-190], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.2e-157], t$95$1, If[LessEqual[z, 7.2e-6], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < 5.5000000000000003e-297 or 6.9999999999999999e-190 < z < 6.1999999999999996e-157

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 88.0%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 87.9%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{y \cdot \frac{\left(1 + \log z\right) - z}{x}}\right) \]

      2. +-commutative 87.9%

         \[\leadsto x \cdot \left(0.5 + y \cdot \frac{\color{blue}{\left(\log z + 1\right)} - z}{x}\right) \]

      3. associate--l+ 87.9%

         \[\leadsto x \cdot \left(0.5 + y \cdot \frac{\color{blue}{\log z + \left(1 - z\right)}}{x}\right) \]

   5. Simplified 87.9%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y \cdot \frac{\log z + \left(1 - z\right)}{x}\right)} \]
   6. Step-by-step derivation

      1. clear-num 87.9%

         \[\leadsto x \cdot \left(0.5 + y \cdot \color{blue}{\frac{1}{\frac{x}{\log z + \left(1 - z\right)}}}\right) \]

      2. un-div-inv 87.9%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{\frac{x}{\log z + \left(1 - z\right)}}}\right) \]

      3. associate-+r- 87.9%

         \[\leadsto x \cdot \left(0.5 + \frac{y}{\frac{x}{\color{blue}{\left(\log z + 1\right) - z}}}\right) \]

      4. +-commutative 87.9%

         \[\leadsto x \cdot \left(0.5 + \frac{y}{\frac{x}{\color{blue}{\left(1 + \log z\right)} - z}}\right) \]

      5. associate--l+ 87.9%

         \[\leadsto x \cdot \left(0.5 + \frac{y}{\frac{x}{\color{blue}{1 + \left(\log z - z\right)}}}\right) \]

   7. Applied egg-rr 87.9%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{\frac{x}{1 + \left(\log z - z\right)}}}\right) \]
   8. Step-by-step derivation

      1. associate-/r/ 88.0%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{x} \cdot \left(1 + \left(\log z - z\right)\right)}\right) \]

   9. Simplified 88.0%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{x} \cdot \left(1 + \left(\log z - z\right)\right)}\right) \]

   10. Taylor expanded in z around inf 74.1%

       \[\leadsto x \cdot \left(0.5 + \frac{y}{x} \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg 74.1%

          \[\leadsto x \cdot \left(0.5 + \frac{y}{x} \cdot \color{blue}{\left(-z\right)}\right) \]

   12. Simplified 74.1%

       \[\leadsto x \cdot \left(0.5 + \frac{y}{x} \cdot \color{blue}{\left(-z\right)}\right) \]
   13. Step-by-step derivation

       1. distribute-rgt-neg-out 74.1%

          \[\leadsto x \cdot \left(0.5 + \color{blue}{\left(-\frac{y}{x} \cdot z\right)}\right) \]

       2. unsub-neg 74.1%

          \[\leadsto x \cdot \color{blue}{\left(0.5 - \frac{y}{x} \cdot z\right)} \]

       3. *-commutative 74.1%

          \[\leadsto x \cdot \left(0.5 - \color{blue}{z \cdot \frac{y}{x}}\right) \]

   14. Applied egg-rr 74.1%

       \[\leadsto x \cdot \color{blue}{\left(0.5 - z \cdot \frac{y}{x}\right)} \]

   if 5.5000000000000003e-297 < z < 1e-264 or 6.1999999999999996e-157 < z < 7.19999999999999967e-6

   1. Initial program 99.7%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 99.7%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   5. Taylor expanded in x around 0 67.8%

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

   6. Taylor expanded in z around 0 65.8%

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

   if 1e-264 < z < 2.2500000000000001e-256

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 2.2500000000000001e-256 < z < 6.9999999999999999e-190

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 99.5%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   4. Applied egg-rr 99.5%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   5. Taylor expanded in x around 0 66.0%

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

   6. Taylor expanded in z around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. distribute-rgt1-in 66.2%

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

      3. +-commutative 66.2%

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

   8. Applied egg-rr 66.2%

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

   if 7.19999999999999967e-6 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right) + x \cdot 0.5} \]

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(1 - z\right) + \log z, x \cdot 0.5\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(1 - z\right) + \log z, x \cdot 0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.7%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{-1 \cdot z}, x \cdot 0.5\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.7%

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

   7. Simplified 99.7%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{-z}, x \cdot 0.5\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{-297}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 10^{-264}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-256}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-190}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ t_1 := x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{if}\;z \leq 1.15 \cdot 10^{-296}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-265}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-256}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))) (t_1 (* x (- 0.5 (* z (/ y x))))))
   (if (<= z 1.15e-296)
     t_1
     (if (<= z 5.6e-265)
       t_0
       (if (<= z 8e-256)
         (* x 0.5)
         (if (<= z 9.2e-189)
           t_0
           (if (<= z 8e-159)
             t_1
             (if (<= z 1.08e-5) t_0 (- (* x 0.5) (* y z))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double t_1 = x * (0.5 - (z * (y / x)));
	double tmp;
	if (z <= 1.15e-296) {
		tmp = t_1;
	} else if (z <= 5.6e-265) {
		tmp = t_0;
	} else if (z <= 8e-256) {
		tmp = x * 0.5;
	} else if (z <= 9.2e-189) {
		tmp = t_0;
	} else if (z <= 8e-159) {
		tmp = t_1;
	} else if (z <= 1.08e-5) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * 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) :: t_1
    real(8) :: tmp
    t_0 = y * (1.0d0 + log(z))
    t_1 = x * (0.5d0 - (z * (y / x)))
    if (z <= 1.15d-296) then
        tmp = t_1
    else if (z <= 5.6d-265) then
        tmp = t_0
    else if (z <= 8d-256) then
        tmp = x * 0.5d0
    else if (z <= 9.2d-189) then
        tmp = t_0
    else if (z <= 8d-159) then
        tmp = t_1
    else if (z <= 1.08d-5) then
        tmp = t_0
    else
        tmp = (x * 0.5d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + Math.log(z));
	double t_1 = x * (0.5 - (z * (y / x)));
	double tmp;
	if (z <= 1.15e-296) {
		tmp = t_1;
	} else if (z <= 5.6e-265) {
		tmp = t_0;
	} else if (z <= 8e-256) {
		tmp = x * 0.5;
	} else if (z <= 9.2e-189) {
		tmp = t_0;
	} else if (z <= 8e-159) {
		tmp = t_1;
	} else if (z <= 1.08e-5) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 + math.log(z))
	t_1 = x * (0.5 - (z * (y / x)))
	tmp = 0
	if z <= 1.15e-296:
		tmp = t_1
	elif z <= 5.6e-265:
		tmp = t_0
	elif z <= 8e-256:
		tmp = x * 0.5
	elif z <= 9.2e-189:
		tmp = t_0
	elif z <= 8e-159:
		tmp = t_1
	elif z <= 1.08e-5:
		tmp = t_0
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	t_1 = Float64(x * Float64(0.5 - Float64(z * Float64(y / x))))
	tmp = 0.0
	if (z <= 1.15e-296)
		tmp = t_1;
	elseif (z <= 5.6e-265)
		tmp = t_0;
	elseif (z <= 8e-256)
		tmp = Float64(x * 0.5);
	elseif (z <= 9.2e-189)
		tmp = t_0;
	elseif (z <= 8e-159)
		tmp = t_1;
	elseif (z <= 1.08e-5)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + log(z));
	t_1 = x * (0.5 - (z * (y / x)));
	tmp = 0.0;
	if (z <= 1.15e-296)
		tmp = t_1;
	elseif (z <= 5.6e-265)
		tmp = t_0;
	elseif (z <= 8e-256)
		tmp = x * 0.5;
	elseif (z <= 9.2e-189)
		tmp = t_0;
	elseif (z <= 8e-159)
		tmp = t_1;
	elseif (z <= 1.08e-5)
		tmp = t_0;
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(0.5 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.15e-296], t$95$1, If[LessEqual[z, 5.6e-265], t$95$0, If[LessEqual[z, 8e-256], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 9.2e-189], t$95$0, If[LessEqual[z, 8e-159], t$95$1, If[LessEqual[z, 1.08e-5], t$95$0, N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.15000000000000002e-296 or 9.1999999999999993e-189 < z < 7.99999999999999991e-159

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 88.0%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 87.9%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{y \cdot \frac{\left(1 + \log z\right) - z}{x}}\right) \]

      2. +-commutative 87.9%

         \[\leadsto x \cdot \left(0.5 + y \cdot \frac{\color{blue}{\left(\log z + 1\right)} - z}{x}\right) \]

      3. associate--l+ 87.9%

         \[\leadsto x \cdot \left(0.5 + y \cdot \frac{\color{blue}{\log z + \left(1 - z\right)}}{x}\right) \]

   5. Simplified 87.9%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y \cdot \frac{\log z + \left(1 - z\right)}{x}\right)} \]
   6. Step-by-step derivation

      1. clear-num 87.9%

         \[\leadsto x \cdot \left(0.5 + y \cdot \color{blue}{\frac{1}{\frac{x}{\log z + \left(1 - z\right)}}}\right) \]

      2. un-div-inv 87.9%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{\frac{x}{\log z + \left(1 - z\right)}}}\right) \]

      3. associate-+r- 87.9%

         \[\leadsto x \cdot \left(0.5 + \frac{y}{\frac{x}{\color{blue}{\left(\log z + 1\right) - z}}}\right) \]

      4. +-commutative 87.9%

         \[\leadsto x \cdot \left(0.5 + \frac{y}{\frac{x}{\color{blue}{\left(1 + \log z\right)} - z}}\right) \]

      5. associate--l+ 87.9%

         \[\leadsto x \cdot \left(0.5 + \frac{y}{\frac{x}{\color{blue}{1 + \left(\log z - z\right)}}}\right) \]

   7. Applied egg-rr 87.9%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{\frac{x}{1 + \left(\log z - z\right)}}}\right) \]
   8. Step-by-step derivation

      1. associate-/r/ 88.0%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{x} \cdot \left(1 + \left(\log z - z\right)\right)}\right) \]

   9. Simplified 88.0%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{x} \cdot \left(1 + \left(\log z - z\right)\right)}\right) \]

   10. Taylor expanded in z around inf 74.1%

       \[\leadsto x \cdot \left(0.5 + \frac{y}{x} \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg 74.1%

          \[\leadsto x \cdot \left(0.5 + \frac{y}{x} \cdot \color{blue}{\left(-z\right)}\right) \]

   12. Simplified 74.1%

       \[\leadsto x \cdot \left(0.5 + \frac{y}{x} \cdot \color{blue}{\left(-z\right)}\right) \]
   13. Step-by-step derivation

       1. distribute-rgt-neg-out 74.1%

          \[\leadsto x \cdot \left(0.5 + \color{blue}{\left(-\frac{y}{x} \cdot z\right)}\right) \]

       2. unsub-neg 74.1%

          \[\leadsto x \cdot \color{blue}{\left(0.5 - \frac{y}{x} \cdot z\right)} \]

       3. *-commutative 74.1%

          \[\leadsto x \cdot \left(0.5 - \color{blue}{z \cdot \frac{y}{x}}\right) \]

   14. Applied egg-rr 74.1%

       \[\leadsto x \cdot \color{blue}{\left(0.5 - z \cdot \frac{y}{x}\right)} \]

   if 1.15000000000000002e-296 < z < 5.60000000000000047e-265 or 7.99999999999999982e-256 < z < 9.1999999999999993e-189 or 7.99999999999999991e-159 < z < 1.07999999999999999e-5

   1. Initial program 99.7%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 99.6%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   4. Applied egg-rr 99.6%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   5. Taylor expanded in x around 0 67.3%

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

   6. Taylor expanded in z around 0 65.9%

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

      1. *-commutative 65.9%

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

      2. distribute-rgt1-in 65.9%

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

      3. +-commutative 65.9%

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

   8. Applied egg-rr 65.9%

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

   if 5.60000000000000047e-265 < z < 7.99999999999999982e-256

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 1.07999999999999999e-5 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 99.6%

      \[\leadsto x \cdot 0.5 + \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.6%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(-1 \cdot y\right) \cdot z} \]

      2. mul-1-neg 99.6%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right)} \cdot z \]

   5. Simplified 99.6%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right) \cdot z} \]
   6. Step-by-step derivation

      1. fma-define 99.6%

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

      2. distribute-lft-neg-out 99.6%

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

      3. add-log-exp 43.9%

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

      4. exp-neg 43.8%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \color{blue}{\left(\frac{1}{e^{y \cdot z}}\right)}\right) \]

      5. *-commutative 43.8%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{e^{\color{blue}{z \cdot y}}}\right)\right) \]

      6. exp-prod 29.5%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{\color{blue}{{\left(e^{z}\right)}^{y}}}\right)\right) \]

      7. add-sqr-sqrt 15.2%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}}}\right)\right) \]

      8. sqrt-unprod 28.5%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(\sqrt{y \cdot y}\right)}}}\right)\right) \]

      9. sqr-neg 28.5%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\left(\sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}\right)}}\right)\right) \]

      10. sqrt-unprod 0.7%

          \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}}}\right)\right) \]

      11. add-sqr-sqrt 2.6%

          \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(-y\right)}}}\right)\right) \]

      12. exp-prod 15.9%

          \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{\color{blue}{e^{z \cdot \left(-y\right)}}}\right)\right) \]

      13. *-commutative 15.9%

          \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{e^{\color{blue}{\left(-y\right) \cdot z}}}\right)\right) \]

      14. neg-log 15.9%

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

      15. add-log-exp 27.7%

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

      16. fma-neg 27.7%

          \[\leadsto \color{blue}{x \cdot 0.5 - \left(-y\right) \cdot z} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-296}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-265}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-256}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-189}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + y \cdot \log z\\ t_1 := x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{if}\;z \leq 8.5 \cdot 10^{-297}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-264}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-255}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-189}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 10^{-156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (* y (log z)))) (t_1 (* x (- 0.5 (* z (/ y x))))))
   (if (<= z 8.5e-297)
     t_1
     (if (<= z 1.6e-264)
       t_0
       (if (<= z 6.2e-255)
         (* x 0.5)
         (if (<= z 1.45e-189)
           (* y (+ 1.0 (log z)))
           (if (<= z 1e-156)
             t_1
             (if (<= z 3.3e-5) t_0 (- (* x 0.5) (* y z))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (y * log(z));
	double t_1 = x * (0.5 - (z * (y / x)));
	double tmp;
	if (z <= 8.5e-297) {
		tmp = t_1;
	} else if (z <= 1.6e-264) {
		tmp = t_0;
	} else if (z <= 6.2e-255) {
		tmp = x * 0.5;
	} else if (z <= 1.45e-189) {
		tmp = y * (1.0 + log(z));
	} else if (z <= 1e-156) {
		tmp = t_1;
	} else if (z <= 3.3e-5) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * 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) :: t_1
    real(8) :: tmp
    t_0 = y + (y * log(z))
    t_1 = x * (0.5d0 - (z * (y / x)))
    if (z <= 8.5d-297) then
        tmp = t_1
    else if (z <= 1.6d-264) then
        tmp = t_0
    else if (z <= 6.2d-255) then
        tmp = x * 0.5d0
    else if (z <= 1.45d-189) then
        tmp = y * (1.0d0 + log(z))
    else if (z <= 1d-156) then
        tmp = t_1
    else if (z <= 3.3d-5) then
        tmp = t_0
    else
        tmp = (x * 0.5d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y + (y * Math.log(z));
	double t_1 = x * (0.5 - (z * (y / x)));
	double tmp;
	if (z <= 8.5e-297) {
		tmp = t_1;
	} else if (z <= 1.6e-264) {
		tmp = t_0;
	} else if (z <= 6.2e-255) {
		tmp = x * 0.5;
	} else if (z <= 1.45e-189) {
		tmp = y * (1.0 + Math.log(z));
	} else if (z <= 1e-156) {
		tmp = t_1;
	} else if (z <= 3.3e-5) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (y * math.log(z))
	t_1 = x * (0.5 - (z * (y / x)))
	tmp = 0
	if z <= 8.5e-297:
		tmp = t_1
	elif z <= 1.6e-264:
		tmp = t_0
	elif z <= 6.2e-255:
		tmp = x * 0.5
	elif z <= 1.45e-189:
		tmp = y * (1.0 + math.log(z))
	elif z <= 1e-156:
		tmp = t_1
	elif z <= 3.3e-5:
		tmp = t_0
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(y * log(z)))
	t_1 = Float64(x * Float64(0.5 - Float64(z * Float64(y / x))))
	tmp = 0.0
	if (z <= 8.5e-297)
		tmp = t_1;
	elseif (z <= 1.6e-264)
		tmp = t_0;
	elseif (z <= 6.2e-255)
		tmp = Float64(x * 0.5);
	elseif (z <= 1.45e-189)
		tmp = Float64(y * Float64(1.0 + log(z)));
	elseif (z <= 1e-156)
		tmp = t_1;
	elseif (z <= 3.3e-5)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (y * log(z));
	t_1 = x * (0.5 - (z * (y / x)));
	tmp = 0.0;
	if (z <= 8.5e-297)
		tmp = t_1;
	elseif (z <= 1.6e-264)
		tmp = t_0;
	elseif (z <= 6.2e-255)
		tmp = x * 0.5;
	elseif (z <= 1.45e-189)
		tmp = y * (1.0 + log(z));
	elseif (z <= 1e-156)
		tmp = t_1;
	elseif (z <= 3.3e-5)
		tmp = t_0;
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(0.5 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 8.5e-297], t$95$1, If[LessEqual[z, 1.6e-264], t$95$0, If[LessEqual[z, 6.2e-255], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 1.45e-189], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e-156], t$95$1, If[LessEqual[z, 3.3e-5], t$95$0, N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < 8.49999999999999991e-297 or 1.45e-189 < z < 1.00000000000000004e-156

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 88.0%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 87.9%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{y \cdot \frac{\left(1 + \log z\right) - z}{x}}\right) \]

      2. +-commutative 87.9%

         \[\leadsto x \cdot \left(0.5 + y \cdot \frac{\color{blue}{\left(\log z + 1\right)} - z}{x}\right) \]

      3. associate--l+ 87.9%

         \[\leadsto x \cdot \left(0.5 + y \cdot \frac{\color{blue}{\log z + \left(1 - z\right)}}{x}\right) \]

   5. Simplified 87.9%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y \cdot \frac{\log z + \left(1 - z\right)}{x}\right)} \]
   6. Step-by-step derivation

      1. clear-num 87.9%

         \[\leadsto x \cdot \left(0.5 + y \cdot \color{blue}{\frac{1}{\frac{x}{\log z + \left(1 - z\right)}}}\right) \]

      2. un-div-inv 87.9%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{\frac{x}{\log z + \left(1 - z\right)}}}\right) \]

      3. associate-+r- 87.9%

         \[\leadsto x \cdot \left(0.5 + \frac{y}{\frac{x}{\color{blue}{\left(\log z + 1\right) - z}}}\right) \]

      4. +-commutative 87.9%

         \[\leadsto x \cdot \left(0.5 + \frac{y}{\frac{x}{\color{blue}{\left(1 + \log z\right)} - z}}\right) \]

      5. associate--l+ 87.9%

         \[\leadsto x \cdot \left(0.5 + \frac{y}{\frac{x}{\color{blue}{1 + \left(\log z - z\right)}}}\right) \]

   7. Applied egg-rr 87.9%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{\frac{x}{1 + \left(\log z - z\right)}}}\right) \]
   8. Step-by-step derivation

      1. associate-/r/ 88.0%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{x} \cdot \left(1 + \left(\log z - z\right)\right)}\right) \]

   9. Simplified 88.0%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{\frac{y}{x} \cdot \left(1 + \left(\log z - z\right)\right)}\right) \]

   10. Taylor expanded in z around inf 74.1%

       \[\leadsto x \cdot \left(0.5 + \frac{y}{x} \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg 74.1%

          \[\leadsto x \cdot \left(0.5 + \frac{y}{x} \cdot \color{blue}{\left(-z\right)}\right) \]

   12. Simplified 74.1%

       \[\leadsto x \cdot \left(0.5 + \frac{y}{x} \cdot \color{blue}{\left(-z\right)}\right) \]
   13. Step-by-step derivation

       1. distribute-rgt-neg-out 74.1%

          \[\leadsto x \cdot \left(0.5 + \color{blue}{\left(-\frac{y}{x} \cdot z\right)}\right) \]

       2. unsub-neg 74.1%

          \[\leadsto x \cdot \color{blue}{\left(0.5 - \frac{y}{x} \cdot z\right)} \]

       3. *-commutative 74.1%

          \[\leadsto x \cdot \left(0.5 - \color{blue}{z \cdot \frac{y}{x}}\right) \]

   14. Applied egg-rr 74.1%

       \[\leadsto x \cdot \color{blue}{\left(0.5 - z \cdot \frac{y}{x}\right)} \]

   if 8.49999999999999991e-297 < z < 1.59999999999999998e-264 or 1.00000000000000004e-156 < z < 3.3000000000000003e-5

   1. Initial program 99.7%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 99.7%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   5. Taylor expanded in x around 0 67.8%

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

   6. Taylor expanded in z around 0 65.8%

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

   if 1.59999999999999998e-264 < z < 6.19999999999999995e-255

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 6.19999999999999995e-255 < z < 1.45e-189

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 99.5%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   4. Applied egg-rr 99.5%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   5. Taylor expanded in x around 0 66.0%

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

   6. Taylor expanded in z around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. distribute-rgt1-in 66.2%

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

      3. +-commutative 66.2%

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

   8. Applied egg-rr 66.2%

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

   if 3.3000000000000003e-5 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 99.6%

      \[\leadsto x \cdot 0.5 + \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.6%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(-1 \cdot y\right) \cdot z} \]

      2. mul-1-neg 99.6%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right)} \cdot z \]

   5. Simplified 99.6%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right) \cdot z} \]
   6. Step-by-step derivation

      1. fma-define 99.6%

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

      2. distribute-lft-neg-out 99.6%

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

      3. add-log-exp 43.9%

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

      4. exp-neg 43.8%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \color{blue}{\left(\frac{1}{e^{y \cdot z}}\right)}\right) \]

      5. *-commutative 43.8%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{e^{\color{blue}{z \cdot y}}}\right)\right) \]

      6. exp-prod 29.5%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{\color{blue}{{\left(e^{z}\right)}^{y}}}\right)\right) \]

      7. add-sqr-sqrt 15.2%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}}}\right)\right) \]

      8. sqrt-unprod 28.5%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(\sqrt{y \cdot y}\right)}}}\right)\right) \]

      9. sqr-neg 28.5%

         \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\left(\sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}\right)}}\right)\right) \]

      10. sqrt-unprod 0.7%

          \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}}}\right)\right) \]

      11. add-sqr-sqrt 2.6%

          \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(-y\right)}}}\right)\right) \]

      12. exp-prod 15.9%

          \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{\color{blue}{e^{z \cdot \left(-y\right)}}}\right)\right) \]

      13. *-commutative 15.9%

          \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{e^{\color{blue}{\left(-y\right) \cdot z}}}\right)\right) \]

      14. neg-log 15.9%

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

      15. add-log-exp 27.7%

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

      16. fma-neg 27.7%

          \[\leadsto \color{blue}{x \cdot 0.5 - \left(-y\right) \cdot z} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{-297}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-264}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-255}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-189}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 10^{-156}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+52} \lor \neg \left(y \leq 1.9 \cdot 10^{-37}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.8e+52) (not (<= y 1.9e-37)))
   (* y (- (+ 1.0 (log z)) z))
   (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.8e+52) || !(y <= 1.9e-37)) {
		tmp = y * ((1.0 + log(z)) - z);
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.8e+52) || !(y <= 1.9e-37))
		tmp = Float64(y * Float64(Float64(1.0 + log(z)) - z));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.8e+52], N[Not[LessEqual[y, 1.9e-37]], $MachinePrecision]], N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e52 or 1.9000000000000002e-37 < y

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 76.0%

      \[\leadsto x \cdot 0.5 + \color{blue}{z \cdot \left(-1 \cdot y + \frac{y \cdot \left(1 + -1 \cdot \log \left(\frac{1}{z}\right)\right)}{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 76.0%

         \[\leadsto x \cdot 0.5 + z \cdot \left(\color{blue}{y \cdot -1} + \frac{y \cdot \left(1 + -1 \cdot \log \left(\frac{1}{z}\right)\right)}{z}\right) \]

      2. associate-/l* 76.8%

         \[\leadsto x \cdot 0.5 + z \cdot \left(y \cdot -1 + \color{blue}{y \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{z}\right)}{z}}\right) \]

      3. distribute-lft-out 76.8%

         \[\leadsto x \cdot 0.5 + z \cdot \color{blue}{\left(y \cdot \left(-1 + \frac{1 + -1 \cdot \log \left(\frac{1}{z}\right)}{z}\right)\right)} \]

      4. mul-1-neg 76.8%

         \[\leadsto x \cdot 0.5 + z \cdot \left(y \cdot \left(-1 + \frac{1 + \color{blue}{\left(-\log \left(\frac{1}{z}\right)\right)}}{z}\right)\right) \]

      5. log-rec 76.8%

         \[\leadsto x \cdot 0.5 + z \cdot \left(y \cdot \left(-1 + \frac{1 + \left(-\color{blue}{\left(-\log z\right)}\right)}{z}\right)\right) \]

      6. remove-double-neg 76.8%

         \[\leadsto x \cdot 0.5 + z \cdot \left(y \cdot \left(-1 + \frac{1 + \color{blue}{\log z}}{z}\right)\right) \]

   5. Simplified 76.8%

      \[\leadsto x \cdot 0.5 + \color{blue}{z \cdot \left(y \cdot \left(-1 + \frac{1 + \log z}{z}\right)\right)} \]

   6. Taylor expanded in z around 0 99.0%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + y \cdot \left(1 + \log z\right)\right)} \]

   7. Taylor expanded in x around 0 88.8%

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

   8. Taylor expanded in y around 0 89.6%

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

      1. associate-+r+ 89.6%

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

      2. mul-1-neg 89.6%

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

      3. sub-neg 89.6%

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

   10. Simplified 89.6%

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

   if -2.8e52 < y < 1.9000000000000002e-37

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right) + x \cdot 0.5} \]

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(1 - z\right) + \log z, x \cdot 0.5\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(1 - z\right) + \log z, x \cdot 0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 89.7%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{-1 \cdot z}, x \cdot 0.5\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 89.7%

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

   7. Simplified 89.7%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+52} \lor \neg \left(y \leq 1.9 \cdot 10^{-37}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.28:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.28) (+ (* x 0.5) (* y (+ 1.0 (log z)))) (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.28)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(1.0 + log(z))));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.28], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 98.7%

      \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(1 + \log z\right)} \]

   if 0.28000000000000003 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right) + x \cdot 0.5} \]

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(1 - z\right) + \log z, x \cdot 0.5\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(1 - z\right) + \log z, x \cdot 0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.7%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{-1 \cdot z}, x \cdot 0.5\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.28:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(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 * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + Math.log(z)));
}

Python

def code(x, y, z):
	return (x * 0.5) + (y * ((1.0 - z) + math.log(z)))

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(y * Float64(Float64(1.0 - z) + log(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.5) + (y * ((1.0 - z) + log(z)));
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
4. Add Preprocessing

Alternative 8: 59.2% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+39}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.3e+39) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.3e+39) {
		tmp = x * 0.5;
	} else {
		tmp = y * -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) :: tmp
    if (z <= 2.3d+39) then
        tmp = x * 0.5d0
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.3e+39) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.3e+39:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.3e+39)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.3e+39)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.3e+39], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.30000000000000012e39

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 49.5%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 2.30000000000000012e39 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 99.2%

      \[\leadsto x \cdot 0.5 + \color{blue}{z \cdot \left(-1 \cdot y + \frac{y \cdot \left(1 + -1 \cdot \log \left(\frac{1}{z}\right)\right)}{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto x \cdot 0.5 + z \cdot \left(\color{blue}{y \cdot -1} + \frac{y \cdot \left(1 + -1 \cdot \log \left(\frac{1}{z}\right)\right)}{z}\right) \]

      2. associate-/l* 100.0%

         \[\leadsto x \cdot 0.5 + z \cdot \left(y \cdot -1 + \color{blue}{y \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{z}\right)}{z}}\right) \]

      3. distribute-lft-out 100.0%

         \[\leadsto x \cdot 0.5 + z \cdot \color{blue}{\left(y \cdot \left(-1 + \frac{1 + -1 \cdot \log \left(\frac{1}{z}\right)}{z}\right)\right)} \]

      4. mul-1-neg 100.0%

         \[\leadsto x \cdot 0.5 + z \cdot \left(y \cdot \left(-1 + \frac{1 + \color{blue}{\left(-\log \left(\frac{1}{z}\right)\right)}}{z}\right)\right) \]

      5. log-rec 100.0%

         \[\leadsto x \cdot 0.5 + z \cdot \left(y \cdot \left(-1 + \frac{1 + \left(-\color{blue}{\left(-\log z\right)}\right)}{z}\right)\right) \]

      6. remove-double-neg 100.0%

         \[\leadsto x \cdot 0.5 + z \cdot \left(y \cdot \left(-1 + \frac{1 + \color{blue}{\log z}}{z}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto x \cdot 0.5 + \color{blue}{z \cdot \left(y \cdot \left(-1 + \frac{1 + \log z}{z}\right)\right)} \]

   6. Taylor expanded in z around 0 99.2%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + y \cdot \left(1 + \log z\right)\right)} \]

   7. Taylor expanded in x around 0 76.9%

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

   8. Taylor expanded in z around inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. mul-1-neg 77.7%

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

   10. Simplified 77.7%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+39}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 - y \cdot z \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return (x * 0.5) - (y * z)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf 75.8%

   \[\leadsto x \cdot 0.5 + \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
4. Step-by-step derivation

   1. associate-*r* 75.8%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-1 \cdot y\right) \cdot z} \]

   2. mul-1-neg 75.8%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right)} \cdot z \]

5. Simplified 75.8%

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right) \cdot z} \]
6. Step-by-step derivation

   1. fma-define 75.8%

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

   2. distribute-lft-neg-out 75.8%

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

   3. add-log-exp 44.8%

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

   4. exp-neg 44.8%

      \[\leadsto \mathsf{fma}\left(x, 0.5, \log \color{blue}{\left(\frac{1}{e^{y \cdot z}}\right)}\right) \]

   5. *-commutative 44.8%

      \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{e^{\color{blue}{z \cdot y}}}\right)\right) \]

   6. exp-prod 36.9%

      \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{\color{blue}{{\left(e^{z}\right)}^{y}}}\right)\right) \]

   7. add-sqr-sqrt 29.1%

      \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}}}\right)\right) \]

   8. sqrt-unprod 36.4%

      \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(\sqrt{y \cdot y}\right)}}}\right)\right) \]

   9. sqr-neg 36.4%

      \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\left(\sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}\right)}}\right)\right) \]

   10. sqrt-unprod 21.1%

       \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}}}\right)\right) \]

   11. add-sqr-sqrt 22.1%

       \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{{\left(e^{z}\right)}^{\color{blue}{\left(-y\right)}}}\right)\right) \]

   12. exp-prod 28.8%

       \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{\color{blue}{e^{z \cdot \left(-y\right)}}}\right)\right) \]

   13. *-commutative 28.8%

       \[\leadsto \mathsf{fma}\left(x, 0.5, \log \left(\frac{1}{e^{\color{blue}{\left(-y\right) \cdot z}}}\right)\right) \]

   14. neg-log 28.8%

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

   15. add-log-exp 35.7%

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

   16. fma-neg 35.7%

       \[\leadsto \color{blue}{x \cdot 0.5 - \left(-y\right) \cdot z} \]

7. Applied egg-rr 75.8%

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

8. Final simplification 75.8%

   \[\leadsto x \cdot 0.5 - y \cdot z \]
9. Add Preprocessing

Alternative 10: 40.3% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x 0.5))

C

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

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 * 0.5d0
end function

Java

public static double code(double x, double y, double z) {
	return x * 0.5;
}

Python

def code(x, y, z):
	return x * 0.5

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 36.6%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

4. Final simplification 36.6%

   \[\leadsto x \cdot 0.5 \]
5. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024080 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

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

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))