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

Percentage Accurate: 99.9% → 99.9%

Time: 9.8s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - z\right) + \log z \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ (- 1.0 z) (log z)) -5e+16)
   (fma y (- z) (* x 0.5))
   (+ (* x 0.5) (* y (+ 1.0 (log z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) + log(z)) <= -5e+16) {
		tmp = fma(y, -z, (x * 0.5));
	} else {
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(1.0 - z) + log(z)) <= -5e+16)
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	else
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(1.0 + log(z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], -5e+16], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -5e16

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   if -5e16 < (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 99.0%

      \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(1 + \log z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ \mathbf{if}\;z \leq 6 \cdot 10^{-216}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{-y}{x} - -0.5\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))))
   (if (<= z 6e-216)
     t_0
     (if (<= z 9e-150)
       (* x (- (* z (/ (- y) x)) -0.5))
       (if (<= z 2.15e-44) t_0 (fma y (- z) (* x 0.5)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double tmp;
	if (z <= 6e-216) {
		tmp = t_0;
	} else if (z <= 9e-150) {
		tmp = x * ((z * (-y / x)) - -0.5);
	} else if (z <= 2.15e-44) {
		tmp = t_0;
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	tmp = 0.0
	if (z <= 6e-216)
		tmp = t_0;
	elseif (z <= 9e-150)
		tmp = Float64(x * Float64(Float64(z * Float64(Float64(-y) / x)) - -0.5));
	elseif (z <= 2.15e-44)
		tmp = t_0;
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 6e-216], t$95$0, If[LessEqual[z, 9e-150], N[(x * N[(N[(z * N[((-y) / x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.15e-44], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 6.00000000000000025e-216 or 9.0000000000000005e-150 < z < 2.15000000000000007e-44

   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. +-commutative 99.8%

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

      2. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.8%

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

   6. Taylor expanded in y around inf 63.6%

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

   if 6.00000000000000025e-216 < z < 9.0000000000000005e-150

   1. Initial program 99.8%

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

   3. Taylor expanded in x around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. *-commutative 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

   5. Simplified 92.9%

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

   6. Taylor expanded in z around inf 74.8%

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

   if 2.15000000000000007e-44 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 97.5%

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

      1. neg-mul-1 97.5%

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

   7. Simplified 97.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{-216}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{-y}{x} - -0.5\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-44}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ \mathbf{if}\;z \leq 5.3 \cdot 10^{-216}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{-y}{x} - -0.5\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))))
   (if (<= z 5.3e-216)
     t_0
     (if (<= z 1.1e-149)
       (* x (- (* z (/ (- y) x)) -0.5))
       (if (<= z 2.3e-45) t_0 (- (* x 0.5) (* y z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double tmp;
	if (z <= 5.3e-216) {
		tmp = t_0;
	} else if (z <= 1.1e-149) {
		tmp = x * ((z * (-y / x)) - -0.5);
	} else if (z <= 2.3e-45) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + log(z))
    if (z <= 5.3d-216) then
        tmp = t_0
    else if (z <= 1.1d-149) then
        tmp = x * ((z * (-y / x)) - (-0.5d0))
    else if (z <= 2.3d-45) then
        tmp = t_0
    else
        tmp = (x * 0.5d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + Math.log(z));
	double tmp;
	if (z <= 5.3e-216) {
		tmp = t_0;
	} else if (z <= 1.1e-149) {
		tmp = x * ((z * (-y / x)) - -0.5);
	} else if (z <= 2.3e-45) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 + math.log(z))
	tmp = 0
	if z <= 5.3e-216:
		tmp = t_0
	elif z <= 1.1e-149:
		tmp = x * ((z * (-y / x)) - -0.5)
	elif z <= 2.3e-45:
		tmp = t_0
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	tmp = 0.0
	if (z <= 5.3e-216)
		tmp = t_0;
	elseif (z <= 1.1e-149)
		tmp = Float64(x * Float64(Float64(z * Float64(Float64(-y) / x)) - -0.5));
	elseif (z <= 2.3e-45)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + log(z));
	tmp = 0.0;
	if (z <= 5.3e-216)
		tmp = t_0;
	elseif (z <= 1.1e-149)
		tmp = x * ((z * (-y / x)) - -0.5);
	elseif (z <= 2.3e-45)
		tmp = t_0;
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.3e-216], t$95$0, If[LessEqual[z, 1.1e-149], N[(x * N[(N[(z * N[((-y) / x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.3e-45], t$95$0, N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.29999999999999977e-216 or 1.0999999999999999e-149 < z < 2.29999999999999992e-45

   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. +-commutative 99.8%

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

      2. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.8%

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

   6. Taylor expanded in y around inf 63.6%

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

   if 5.29999999999999977e-216 < z < 1.0999999999999999e-149

   1. Initial program 99.8%

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

   3. Taylor expanded in x around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. *-commutative 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

   5. Simplified 92.9%

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

   6. Taylor expanded in z around inf 74.8%

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

   if 2.29999999999999992e-45 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 97.5%

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

      1. associate-*r* 97.5%

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

      2. mul-1-neg 97.5%

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

   5. Simplified 97.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.3 \cdot 10^{-216}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{-y}{x} - -0.5\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-45}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{-9} \lor \neg \left(y \leq 2.6 \cdot 10^{+79}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.75e-9) (not (<= y 2.6e+79)))
   (* y (- (+ 1.0 (log z)) z))
   (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.75e-9) || !(y <= 2.6e+79)) {
		tmp = y * ((1.0 + log(z)) - z);
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.75e-9) || !(y <= 2.6e+79))
		tmp = Float64(y * Float64(Float64(1.0 + log(z)) - z));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.75e-9], N[Not[LessEqual[y, 2.6e+79]], $MachinePrecision]], N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7499999999999998e-9 or 2.60000000000000015e79 < 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. +-commutative 99.8%

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

      2. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. fma-undefine 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-in 98.7%

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

      4. associate-+r+ 98.7%

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

      5. *-commutative 98.7%

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

      6. fma-define 98.7%

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

      7. *-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in y around inf 92.0%

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

   if -2.7499999999999998e-9 < y < 2.60000000000000015e79

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 86.0%

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

      1. neg-mul-1 86.0%

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

   7. Simplified 86.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{-9} \lor \neg \left(y \leq 2.6 \cdot 10^{+79}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 7: 60.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{+15}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 7e+15) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 7e+15) {
		tmp = x * 0.5;
	} else {
		tmp = 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 <= 7d+15) then
        tmp = x * 0.5d0
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 7e+15) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 7e+15:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 7e+15)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 7e+15)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 7e+15], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7e15

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 50.8%

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

   if 7e15 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 99.2%

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

      4. associate-+r+ 99.2%

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

      5. *-commutative 99.2%

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

      6. fma-define 99.2%

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

      7. *-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in z around inf 68.7%

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

      1. mul-1-neg 68.7%

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

      2. distribute-rgt-neg-out 68.7%

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

   9. Simplified 68.7%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{+15}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf 75.2%

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

   1. associate-*r* 75.2%

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

   2. mul-1-neg 75.2%

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

5. Simplified 75.2%

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

6. Final simplification 75.2%

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

Alternative 9: 39.5% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 42.3%

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

4. Final simplification 42.3%

   \[\leadsto x \cdot 0.5 \]
5. 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 2024121 
(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)))))