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

Percentage Accurate: 99.9% → 99.9%

Time: 10.1s

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. Final simplification 99.9%

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

Alternative 2: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ t_1 := x \cdot 0.5 - y \cdot z\\ \mathbf{if}\;z \leq 5.2 \cdot 10^{-151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))) (t_1 (- (* x 0.5) (* y z))))
   (if (<= z 5.2e-151)
     t_0
     (if (<= z 6.4e-123)
       t_1
       (if (<= z 2.25e-115)
         t_0
         (if (<= z 3.5e-64)
           t_1
           (if (<= z 5.2e-11) 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 t_1 = (x * 0.5) - (y * z);
	double tmp;
	if (z <= 5.2e-151) {
		tmp = t_0;
	} else if (z <= 6.4e-123) {
		tmp = t_1;
	} else if (z <= 2.25e-115) {
		tmp = t_0;
	} else if (z <= 3.5e-64) {
		tmp = t_1;
	} else if (z <= 5.2e-11) {
		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)))
	t_1 = Float64(Float64(x * 0.5) - Float64(y * z))
	tmp = 0.0
	if (z <= 5.2e-151)
		tmp = t_0;
	elseif (z <= 6.4e-123)
		tmp = t_1;
	elseif (z <= 2.25e-115)
		tmp = t_0;
	elseif (z <= 3.5e-64)
		tmp = t_1;
	elseif (z <= 5.2e-11)
		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]}, Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.2e-151], t$95$0, If[LessEqual[z, 6.4e-123], t$95$1, If[LessEqual[z, 2.25e-115], t$95$0, If[LessEqual[z, 3.5e-64], t$95$1, If[LessEqual[z, 5.2e-11], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.2000000000000001e-151 or 6.39999999999999957e-123 < z < 2.25000000000000011e-115 or 3.5000000000000003e-64 < z < 5.2000000000000001e-11

   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 99.4%

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

   4. Taylor expanded in x around 0 61.7%

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

   if 5.2000000000000001e-151 < z < 6.39999999999999957e-123 or 2.25000000000000011e-115 < z < 3.5000000000000003e-64

   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 68.3%

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

      1. associate-*r* 68.3%

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

      2. mul-1-neg 68.3%

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

   5. Simplified 68.3%

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

   if 5.2000000000000001e-11 < 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 97.9%

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

      1. mul-1-neg 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-151}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-123}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-115}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-64}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{-152} \lor \neg \left(z \leq 5.5 \cdot 10^{-123}\right) \land \left(z \leq 2.25 \cdot 10^{-115} \lor \neg \left(z \leq 1.25 \cdot 10^{-64}\right) \land z \leq 4.4 \cdot 10^{-11}\right):\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 3.4e-152)
         (and (not (<= z 5.5e-123))
              (or (<= z 2.25e-115)
                  (and (not (<= z 1.25e-64)) (<= z 4.4e-11)))))
   (* y (+ 1.0 (log z)))
   (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 3.4e-152) || (!(z <= 5.5e-123) && ((z <= 2.25e-115) || (!(z <= 1.25e-64) && (z <= 4.4e-11))))) {
		tmp = y * (1.0 + log(z));
	} 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) :: tmp
    if ((z <= 3.4d-152) .or. (.not. (z <= 5.5d-123)) .and. (z <= 2.25d-115) .or. (.not. (z <= 1.25d-64)) .and. (z <= 4.4d-11)) then
        tmp = y * (1.0d0 + log(z))
    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 tmp;
	if ((z <= 3.4e-152) || (!(z <= 5.5e-123) && ((z <= 2.25e-115) || (!(z <= 1.25e-64) && (z <= 4.4e-11))))) {
		tmp = y * (1.0 + Math.log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 3.4e-152) or (not (z <= 5.5e-123) and ((z <= 2.25e-115) or (not (z <= 1.25e-64) and (z <= 4.4e-11)))):
		tmp = y * (1.0 + math.log(z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 3.4e-152) || (!(z <= 5.5e-123) && ((z <= 2.25e-115) || (!(z <= 1.25e-64) && (z <= 4.4e-11)))))
		tmp = Float64(y * Float64(1.0 + log(z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 3.4e-152) || (~((z <= 5.5e-123)) && ((z <= 2.25e-115) || (~((z <= 1.25e-64)) && (z <= 4.4e-11)))))
		tmp = y * (1.0 + log(z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 3.4e-152], And[N[Not[LessEqual[z, 5.5e-123]], $MachinePrecision], Or[LessEqual[z, 2.25e-115], And[N[Not[LessEqual[z, 1.25e-64]], $MachinePrecision], LessEqual[z, 4.4e-11]]]]], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.39999999999999984e-152 or 5.5e-123 < z < 2.25000000000000011e-115 or 1.25000000000000008e-64 < z < 4.4000000000000003e-11

   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 99.4%

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

   4. Taylor expanded in x around 0 61.7%

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

   if 3.39999999999999984e-152 < z < 5.5e-123 or 2.25000000000000011e-115 < z < 1.25000000000000008e-64 or 4.4000000000000003e-11 < z

   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 92.9%

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

      1. associate-*r* 92.9%

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

      2. mul-1-neg 92.9%

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

   5. Simplified 92.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{-152} \lor \neg \left(z \leq 5.5 \cdot 10^{-123}\right) \land \left(z \leq 2.25 \cdot 10^{-115} \lor \neg \left(z \leq 1.25 \cdot 10^{-64}\right) \land z \leq 4.4 \cdot 10^{-11}\right):\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -4 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{-21}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -4e-101)
   (fma y (- z) (* x 0.5))
   (if (<= (* x 0.5) 1e-21)
     (* y (- (+ 1.0 (log z)) z))
     (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -4e-101) {
		tmp = fma(y, -z, (x * 0.5));
	} else if ((x * 0.5) <= 1e-21) {
		tmp = y * ((1.0 + log(z)) - z);
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -4e-101)
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	elseif (Float64(x * 0.5) <= 1e-21)
		tmp = Float64(y * Float64(Float64(1.0 + log(z)) - z));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -4e-101], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 1e-21], N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -4.00000000000000021e-101

   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 82.4%

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

      1. mul-1-neg 82.4%

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

   7. Simplified 82.4%

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

   if -4.00000000000000021e-101 < (*.f64 x #s(literal 1/2 binary64)) < 9.99999999999999908e-22

   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 68.8%

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

      1. mul-1-neg 68.8%

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

      2. distribute-rgt-neg-in 68.8%

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

      3. fma-neg 68.8%

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

      4. *-commutative 68.8%

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

      5. +-commutative 68.8%

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

      6. associate--l+ 68.8%

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

      7. +-commutative 68.8%

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

      8. associate-/l* 68.7%

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

      9. +-commutative 68.7%

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

      10. metadata-eval 68.7%

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

   5. Simplified 68.7%

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

      1. clear-num 68.6%

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

      2. un-div-inv 68.7%

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

   7. Applied egg-rr 68.7%

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

   8. Taylor expanded in x around 0 92.6%

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

   if 9.99999999999999908e-22 < (*.f64 x #s(literal 1/2 binary64))

   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 88.7%

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

      1. associate-*r* 88.7%

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

      2. mul-1-neg 88.7%

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

   5. Simplified 88.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -4 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{-21}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.0245:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.0245)
   (+ (* x 0.5) (* y (+ 1.0 (log z))))
   (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.0245) {
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.0245)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(1.0 + log(z))));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.0245], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.024500000000000001

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 98.7%

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

   if 0.024500000000000001 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 99.3%

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

      1. mul-1-neg 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0245:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 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. Final simplification 99.9%

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

Alternative 7: 55.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-41} \lor \neg \left(x \leq 2 \cdot 10^{-21}\right):\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.8e-41) (not (<= x 2e-21))) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.8e-41) || !(x <= 2e-21)) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8.8d-41)) .or. (.not. (x <= 2d-21))) then
        tmp = x * 0.5d0
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.8e-41) || !(x <= 2e-21)) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.8e-41) or not (x <= 2e-21):
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.8e-41) || !(x <= 2e-21))
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.8e-41) || ~((x <= 2e-21)))
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.8e-41], N[Not[LessEqual[x, 2e-21]], $MachinePrecision]], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.7999999999999999e-41 or 1.99999999999999982e-21 < x

   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 67.8%

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

   if -8.7999999999999999e-41 < x < 1.99999999999999982e-21

   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 70.6%

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

      1. mul-1-neg 70.6%

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

      2. distribute-rgt-neg-in 70.6%

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

      3. fma-neg 70.6%

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

      4. *-commutative 70.6%

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

      5. +-commutative 70.6%

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

      6. associate--l+ 70.6%

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

      7. +-commutative 70.6%

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

      8. associate-/l* 70.5%

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

      9. +-commutative 70.5%

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

      10. metadata-eval 70.5%

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

   5. Simplified 70.5%

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

      1. clear-num 70.5%

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

      2. un-div-inv 70.5%

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

   7. Applied egg-rr 70.5%

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

   8. Taylor expanded in z around inf 49.4%

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

      1. mul-1-neg 49.4%

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

      2. distribute-rgt-neg-in 49.4%

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

   10. Simplified 49.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-41} \lor \neg \left(x \leq 2 \cdot 10^{-21}\right):\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% 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 72.6%

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

   1. associate-*r* 72.6%

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

   2. mul-1-neg 72.6%

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

5. Simplified 72.6%

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

6. Final simplification 72.6%

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

Alternative 9: 41.5% 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 41.2%

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

4. Final simplification 41.2%

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

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024078 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

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

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