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

Percentage Accurate: 99.9% → 99.9%

Time: 13.1s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] + 1.0), $MachinePrecision] * y + 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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

   5. associate-+r- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

   9. log-lowering-log.f64 99.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

3. Simplified 99.9%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. fma-define N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\left(\left(1 + \log z\right) - z\right), \color{blue}{y}, \left(x \cdot \frac{1}{2}\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(1 + \left(\log z - z\right)\right), y, \left(x \cdot \frac{1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(1, \left(\log z - z\right)\right), y, \left(x \cdot \frac{1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\log z, z\right)\right), y, \left(x \cdot \frac{1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), z\right)\right), y, \left(x \cdot \frac{1}{2}\right)\right) \]

   9. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), z\right)\right), y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log z - z\\ \mathbf{if}\;y \leq -1.28 \cdot 10^{+157}:\\ \;\;\;\;y + t\_0 \cdot y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-70}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t\_0 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log z) z)))
   (if (<= y -1.28e+157)
     (+ y (* t_0 y))
     (if (<= y 1.3e-70) (- (* x 0.5) (* z y)) (* y (+ t_0 1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = log(z) - z;
	double tmp;
	if (y <= -1.28e+157) {
		tmp = y + (t_0 * y);
	} else if (y <= 1.3e-70) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = y * (t_0 + 1.0);
	}
	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 = log(z) - z
    if (y <= (-1.28d+157)) then
        tmp = y + (t_0 * y)
    else if (y <= 1.3d-70) then
        tmp = (x * 0.5d0) - (z * y)
    else
        tmp = y * (t_0 + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(z) - z;
	double tmp;
	if (y <= -1.28e+157) {
		tmp = y + (t_0 * y);
	} else if (y <= 1.3e-70) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = y * (t_0 + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(z) - z
	tmp = 0
	if y <= -1.28e+157:
		tmp = y + (t_0 * y)
	elif y <= 1.3e-70:
		tmp = (x * 0.5) - (z * y)
	else:
		tmp = y * (t_0 + 1.0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(z) - z)
	tmp = 0.0
	if (y <= -1.28e+157)
		tmp = Float64(y + Float64(t_0 * y));
	elseif (y <= 1.3e-70)
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	else
		tmp = Float64(y * Float64(t_0 + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(z) - z;
	tmp = 0.0;
	if (y <= -1.28e+157)
		tmp = y + (t_0 * y);
	elseif (y <= 1.3e-70)
		tmp = (x * 0.5) - (z * y);
	else
		tmp = y * (t_0 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, -1.28e+157], N[(y + N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-70], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.28000000000000001e157

   1. Initial program 99.7%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

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

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-mul-1 N/A

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

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

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

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

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

      16. log-lowering-log.f64 99.7%

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

   7. Simplified 99.7%

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

      1. associate--r- N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r- N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. /-rgt-identity N/A

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

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

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

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

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

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

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

      14. log-lowering-log.f64 99.7%

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

   9. Applied egg-rr 99.7%

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

   if -1.28000000000000001e157 < y < 1.30000000000000001e-70

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 100.0%

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

      1. flip-- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(y \cdot \frac{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}{\color{blue}{\left(1 + \log z\right) + z}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(y \cdot \frac{1}{\color{blue}{\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}}}\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\frac{y}{\color{blue}{\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}\right)}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \left(\frac{1}{\color{blue}{\frac{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}{\left(1 + \log z\right) + z}}}\right)\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \left(\frac{1}{\left(1 + \log z\right) - \color{blue}{z}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(1 + \log z\right) - z\right)}\right)\right)\right) \]

      8. associate--l+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \left(1 + \color{blue}{\left(\log z - z\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\log z - z\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\log z, \color{blue}{z}\right)\right)\right)\right)\right) \]

      11. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), z\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{-1}{z}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 85.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(-1, \color{blue}{z}\right)\right)\right) \]

   9. Simplified 85.0%

      \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{\left(y \cdot z\right)}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\color{blue}{y} \cdot z\right)\right) \]

       6. *-lowering-*.f64 85.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   12. Simplified 85.1%

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

   if 1.30000000000000001e-70 < y

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-mul-1 N/A

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

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

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

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

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

      16. log-lowering-log.f64 88.6%

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

   7. Simplified 88.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.28 \cdot 10^{+157}:\\ \;\;\;\;y + \left(\log z - z\right) \cdot y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-70}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(\log z - z\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\left(\log z - z\right) + 1\right)\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-71}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ (- (log z) z) 1.0))))
   (if (<= y -1.25e+157) t_0 (if (<= y 4.5e-71) (- (* x 0.5) (* z y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * ((log(z) - z) + 1.0);
	double tmp;
	if (y <= -1.25e+157) {
		tmp = t_0;
	} else if (y <= 4.5e-71) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = t_0;
	}
	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 * ((log(z) - z) + 1.0d0)
    if (y <= (-1.25d+157)) then
        tmp = t_0
    else if (y <= 4.5d-71) then
        tmp = (x * 0.5d0) - (z * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * ((Math.log(z) - z) + 1.0);
	double tmp;
	if (y <= -1.25e+157) {
		tmp = t_0;
	} else if (y <= 4.5e-71) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * ((math.log(z) - z) + 1.0)
	tmp = 0
	if y <= -1.25e+157:
		tmp = t_0
	elif y <= 4.5e-71:
		tmp = (x * 0.5) - (z * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(Float64(log(z) - z) + 1.0))
	tmp = 0.0
	if (y <= -1.25e+157)
		tmp = t_0;
	elseif (y <= 4.5e-71)
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * ((log(z) - z) + 1.0);
	tmp = 0.0;
	if (y <= -1.25e+157)
		tmp = t_0;
	elseif (y <= 4.5e-71)
		tmp = (x * 0.5) - (z * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e+157], t$95$0, If[LessEqual[y, 4.5e-71], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.24999999999999994e157 or 4.5000000000000002e-71 < y

   1. Initial program 99.8%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 99.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-mul-1 N/A

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

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

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

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

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

      16. log-lowering-log.f64 90.8%

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

   7. Simplified 90.8%

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

   if -1.24999999999999994e157 < y < 4.5000000000000002e-71

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 100.0%

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

      1. flip-- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(y \cdot \frac{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}{\color{blue}{\left(1 + \log z\right) + z}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(y \cdot \frac{1}{\color{blue}{\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}}}\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\frac{y}{\color{blue}{\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}\right)}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \left(\frac{1}{\color{blue}{\frac{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}{\left(1 + \log z\right) + z}}}\right)\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \left(\frac{1}{\left(1 + \log z\right) - \color{blue}{z}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(1 + \log z\right) - z\right)}\right)\right)\right) \]

      8. associate--l+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \left(1 + \color{blue}{\left(\log z - z\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\log z - z\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\log z, \color{blue}{z}\right)\right)\right)\right)\right) \]

      11. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), z\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{-1}{z}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 85.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(-1, \color{blue}{z}\right)\right)\right) \]

   9. Simplified 85.0%

      \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{\left(y \cdot z\right)}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\color{blue}{y} \cdot z\right)\right) \]

       6. *-lowering-*.f64 85.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   12. Simplified 85.1%

       \[\leadsto \color{blue}{0.5 \cdot x - y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+157}:\\ \;\;\;\;y \cdot \left(\left(\log z - z\right) + 1\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-71}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(\log z - z\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(\log z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 9e-5) (+ (* x 0.5) (* y (+ (log z) 1.0))) (- (* x 0.5) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 9e-5) {
		tmp = (x * 0.5) + (y * (log(z) + 1.0));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	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 <= 9d-5) then
        tmp = (x * 0.5d0) + (y * (log(z) + 1.0d0))
    else
        tmp = (x * 0.5d0) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 9e-5) {
		tmp = (x * 0.5) + (y * (Math.log(z) + 1.0));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 9e-5:
		tmp = (x * 0.5) + (y * (math.log(z) + 1.0))
	else:
		tmp = (x * 0.5) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 9e-5)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(log(z) + 1.0)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 9e-5)
		tmp = (x * 0.5) + (y * (log(z) + 1.0));
	else
		tmp = (x * 0.5) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 9e-5], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(N[Log[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.00000000000000057e-5

   1. Initial program 99.8%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 99.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(1 + \log z\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \log z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\log z}\right)\right)\right) \]

      3. log-lowering-log.f64 99.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right)\right)\right) \]

   7. Simplified 99.2%

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

   if 9.00000000000000057e-5 < 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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 100.0%

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

      1. flip-- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(y \cdot \frac{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}{\color{blue}{\left(1 + \log z\right) + z}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(y \cdot \frac{1}{\color{blue}{\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}}}\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\frac{y}{\color{blue}{\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}\right)}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \left(\frac{1}{\color{blue}{\frac{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}{\left(1 + \log z\right) + z}}}\right)\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \left(\frac{1}{\left(1 + \log z\right) - \color{blue}{z}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(1 + \log z\right) - z\right)}\right)\right)\right) \]

      8. associate--l+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \left(1 + \color{blue}{\left(\log z - z\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\log z - z\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\log z, \color{blue}{z}\right)\right)\right)\right)\right) \]

      11. log-lowering-log.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), z\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{-1}{z}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 99.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(-1, \color{blue}{z}\right)\right)\right) \]

   9. Simplified 99.2%

      \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{\left(y \cdot z\right)}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\color{blue}{y} \cdot z\right)\right) \]

       6. *-lowering-*.f64 99.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   12. Simplified 99.4%

       \[\leadsto \color{blue}{0.5 \cdot x - y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(\log z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 6: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{-250}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.8e-250) (* y (+ (log z) 1.0)) (- (* x 0.5) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8e-250) {
		tmp = y * (log(z) + 1.0);
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	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 <= 1.8d-250) then
        tmp = y * (log(z) + 1.0d0)
    else
        tmp = (x * 0.5d0) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8e-250) {
		tmp = y * (Math.log(z) + 1.0);
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.8e-250:
		tmp = y * (math.log(z) + 1.0)
	else:
		tmp = (x * 0.5) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.8e-250)
		tmp = Float64(y * Float64(log(z) + 1.0));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.8e-250)
		tmp = y * (log(z) + 1.0);
	else
		tmp = (x * 0.5) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.8e-250], N[(y * N[(N[Log[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.79999999999999991e-250

   1. Initial program 99.7%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

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

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-mul-1 N/A

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

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

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

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

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

      16. log-lowering-log.f64 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in z around 0

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

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

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

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

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

      3. log-lowering-log.f64 63.6%

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

   10. Simplified 63.6%

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

   if 1.79999999999999991e-250 < z

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 99.9%

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

      1. flip-- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(y \cdot \frac{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}{\color{blue}{\left(1 + \log z\right) + z}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(y \cdot \frac{1}{\color{blue}{\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}}}\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\frac{y}{\color{blue}{\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}\right)}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \left(\frac{1}{\color{blue}{\frac{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}{\left(1 + \log z\right) + z}}}\right)\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \left(\frac{1}{\left(1 + \log z\right) - \color{blue}{z}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(1 + \log z\right) - z\right)}\right)\right)\right) \]

      8. associate--l+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \left(1 + \color{blue}{\left(\log z - z\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\log z - z\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\log z, \color{blue}{z}\right)\right)\right)\right)\right) \]

      11. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), z\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{-1}{z}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 79.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(-1, \color{blue}{z}\right)\right)\right) \]

   9. Simplified 79.8%

      \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{\left(y \cdot z\right)}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\color{blue}{y} \cdot z\right)\right) \]

       6. *-lowering-*.f64 79.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   12. Simplified 79.9%

       \[\leadsto \color{blue}{0.5 \cdot x - y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{-250}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.4% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{+50}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.4e+50) (* x 0.5) (* y (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.4e+50) {
		tmp = x * 0.5;
	} else {
		tmp = y * (1.0 - 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 <= 1.4d+50) then
        tmp = x * 0.5d0
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.4e+50) {
		tmp = x * 0.5;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.4e+50:
		tmp = x * 0.5
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.4e+50)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.4e+50)
		tmp = x * 0.5;
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.4e+50], N[(x * 0.5), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.3999999999999999e50

   1. Initial program 99.8%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 99.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 52.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

   7. Simplified 52.2%

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

   if 1.3999999999999999e50 < 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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

      9. log-lowering-log.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-mul-1 N/A

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

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

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

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

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

      16. log-lowering-log.f64 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 87.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{+50}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.4% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

   5. associate-+r- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

   9. log-lowering-log.f64 99.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

3. Simplified 99.9%

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

   1. flip-- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(y \cdot \frac{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}{\color{blue}{\left(1 + \log z\right) + z}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(y \cdot \frac{1}{\color{blue}{\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\frac{y}{\color{blue}{\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\left(1 + \log z\right) + z}{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}\right)}\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \left(\frac{1}{\color{blue}{\frac{\left(1 + \log z\right) \cdot \left(1 + \log z\right) - z \cdot z}{\left(1 + \log z\right) + z}}}\right)\right)\right) \]

   6. flip-- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \left(\frac{1}{\left(1 + \log z\right) - \color{blue}{z}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(1 + \log z\right) - z\right)}\right)\right)\right) \]

   8. associate--l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \left(1 + \color{blue}{\left(\log z - z\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\log z - z\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\log z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   11. log-lowering-log.f64 99.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), z\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Taylor expanded in z around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{-1}{z}\right)}\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 75.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(-1, \color{blue}{z}\right)\right)\right) \]

9. Simplified 75.0%

   \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. unsub-neg N/A

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

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

       \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{\left(y \cdot z\right)}\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\color{blue}{y} \cdot z\right)\right) \]

    6. *-lowering-*.f64 75.1%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

12. Simplified 75.1%

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

13. Final simplification 75.1%

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

Alternative 9: 39.8% 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. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

   5. associate-+r- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

   9. log-lowering-log.f64 99.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 36.6%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

7. Simplified 36.6%

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

8. Final simplification 36.6%

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

Alternative 10: 1.9% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \color{blue}{\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\color{blue}{y} \cdot \left(\left(1 - z\right) + \log z\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - z\right) + \log z\right)}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\log z + \color{blue}{\left(1 - z\right)}\right)\right)\right) \]

   5. associate-+r- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \left(\left(\log z + 1\right) - \color{blue}{z}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\log z + 1\right), \color{blue}{z}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(1 + \log z\right), z\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \log z\right), z\right)\right)\right) \]

   9. log-lowering-log.f64 99.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{log.f64}\left(z\right)\right), z\right)\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in x around 0

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

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

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

   2. associate--l+ N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. *-lft-identity N/A

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

   6. metadata-eval N/A

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

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

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

   8. neg-mul-1 N/A

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

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

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

   10. mul-1-neg N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. remove-double-neg N/A

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

   14. sub-neg N/A

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

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

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

   16. log-lowering-log.f64 64.0%

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

7. Simplified 64.0%

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

8. Taylor expanded in z around inf

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

10. Simplified 39.7%

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

11. Taylor expanded in z around 0

    \[\leadsto \color{blue}{y} \]
12. Step-by-step derivation

13. Simplified 1.8%

    \[\leadsto \color{blue}{y} \]
14. Add Preprocessing

Developer Target 1: 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 2024145 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

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

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