Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 6.8s

Alternatives: 9

Speedup: 2.4×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a - \frac{1}{3}\\ t\_0 \cdot \left(1 + \frac{1}{\sqrt{9 \cdot t\_0}} \cdot rand\right) \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (- a (/ 1.0 3.0))))
   (* t_0 (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 t_0))) rand)))))

C

double code(double a, double rand) {
	double t_0 = a - (1.0 / 3.0);
	return t_0 * (1.0 + ((1.0 / sqrt((9.0 * t_0))) * rand));
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    t_0 = a - (1.0d0 / 3.0d0)
    code = t_0 * (1.0d0 + ((1.0d0 / sqrt((9.0d0 * t_0))) * rand))
end function

Java

public static double code(double a, double rand) {
	double t_0 = a - (1.0 / 3.0);
	return t_0 * (1.0 + ((1.0 / Math.sqrt((9.0 * t_0))) * rand));
}

Python

def code(a, rand):
	t_0 = a - (1.0 / 3.0)
	return t_0 * (1.0 + ((1.0 / math.sqrt((9.0 * t_0))) * rand))

Julia

function code(a, rand)
	t_0 = Float64(a - Float64(1.0 / 3.0))
	return Float64(t_0 * Float64(1.0 + Float64(Float64(1.0 / sqrt(Float64(9.0 * t_0))) * rand)))
end

MATLAB

function tmp = code(a, rand)
	t_0 = a - (1.0 / 3.0);
	tmp = t_0 * (1.0 + ((1.0 / sqrt((9.0 * t_0))) * rand));
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(1.0 + N[(N[(1.0 / N[Sqrt[N[(9.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a - \frac{1}{3}\\ t\_0 \cdot \left(1 + \frac{1}{\sqrt{9 \cdot t\_0}} \cdot rand\right) \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (- a (/ 1.0 3.0))))
   (* t_0 (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 t_0))) rand)))))

C

double code(double a, double rand) {
	double t_0 = a - (1.0 / 3.0);
	return t_0 * (1.0 + ((1.0 / sqrt((9.0 * t_0))) * rand));
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    t_0 = a - (1.0d0 / 3.0d0)
    code = t_0 * (1.0d0 + ((1.0d0 / sqrt((9.0d0 * t_0))) * rand))
end function

Java

public static double code(double a, double rand) {
	double t_0 = a - (1.0 / 3.0);
	return t_0 * (1.0 + ((1.0 / Math.sqrt((9.0 * t_0))) * rand));
}

Python

def code(a, rand):
	t_0 = a - (1.0 / 3.0)
	return t_0 * (1.0 + ((1.0 / math.sqrt((9.0 * t_0))) * rand))

Julia

function code(a, rand)
	t_0 = Float64(a - Float64(1.0 / 3.0))
	return Float64(t_0 * Float64(1.0 + Float64(Float64(1.0 / sqrt(Float64(9.0 * t_0))) * rand)))
end

MATLAB

function tmp = code(a, rand)
	t_0 = a - (1.0 / 3.0);
	tmp = t_0 * (1.0 + ((1.0 / sqrt((9.0 * t_0))) * rand));
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(1.0 + N[(N[(1.0 / N[Sqrt[N[(9.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\frac{rand}{3}}{\sqrt{a - 0.3333333333333333}}, a - 0.3333333333333333, a - 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (fma
  (/ (/ rand 3.0) (sqrt (- a 0.3333333333333333)))
  (- a 0.3333333333333333)
  (- a 0.3333333333333333)))

C

double code(double a, double rand) {
	return fma(((rand / 3.0) / sqrt((a - 0.3333333333333333))), (a - 0.3333333333333333), (a - 0.3333333333333333));
}

Julia

function code(a, rand)
	return fma(Float64(Float64(rand / 3.0) / sqrt(Float64(a - 0.3333333333333333))), Float64(a - 0.3333333333333333), Float64(a - 0.3333333333333333))
end

Wolfram

code[a_, rand_] := N[(N[(N[(rand / 3.0), $MachinePrecision] / N[Sqrt[N[(a - 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a - 0.3333333333333333), $MachinePrecision] + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)} \]

   3. +-commutative N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand + 1\right)} \]

   4. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right) + 1 \cdot \left(a - \frac{1}{3}\right)} \]

   5. *-lft-identity N/A

      \[\leadsto \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right) + \color{blue}{\left(a - \frac{1}{3}\right)} \]

   6. lower-fma.f64 99.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand, a - \frac{1}{3}, a - \frac{1}{3}\right)} \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{rand}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}, a - 0.3333333333333333, a - 0.3333333333333333\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{rand}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{rand}{\color{blue}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   3. lift-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{rand}{\sqrt{\color{blue}{a \cdot 9 + -3}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{rand}{\sqrt{a \cdot 9 + \color{blue}{\frac{-1}{3} \cdot 9}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{rand}{\sqrt{a \cdot 9 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot 9}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto \mathsf{fma}\left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{rand}{\sqrt{9 \cdot \color{blue}{\left(a - \frac{1}{3}\right)}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   8. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{rand}{\sqrt{9 \cdot \color{blue}{\left(a - \frac{1}{3}\right)}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   9. sqrt-prod N/A

      \[\leadsto \mathsf{fma}\left(\frac{rand}{\color{blue}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{rand}{\color{blue}{3} \cdot \sqrt{a - \frac{1}{3}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{rand}{3 \cdot \color{blue}{\sqrt{a - \frac{1}{3}}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   12. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{rand}{3}}{\sqrt{a - \frac{1}{3}}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{rand}{3}}{\sqrt{a - \frac{1}{3}}}}, a - \frac{1}{3}, a - \frac{1}{3}\right) \]

   14. lower-/.f64 99.9

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{rand}{3}}}{\sqrt{a - 0.3333333333333333}}, a - 0.3333333333333333, a - 0.3333333333333333\right) \]

6. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{rand}{3}}{\sqrt{a - 0.3333333333333333}}}, a - 0.3333333333333333, a - 0.3333333333333333\right) \]
7. Add Preprocessing

Alternative 2: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{rand}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}, a - 0.3333333333333333, a - 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (fma
  (/ rand (sqrt (fma a 9.0 -3.0)))
  (- a 0.3333333333333333)
  (- a 0.3333333333333333)))

C

double code(double a, double rand) {
	return fma((rand / sqrt(fma(a, 9.0, -3.0))), (a - 0.3333333333333333), (a - 0.3333333333333333));
}

Julia

function code(a, rand)
	return fma(Float64(rand / sqrt(fma(a, 9.0, -3.0))), Float64(a - 0.3333333333333333), Float64(a - 0.3333333333333333))
end

Wolfram

code[a_, rand_] := N[(N[(rand / N[Sqrt[N[(a * 9.0 + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a - 0.3333333333333333), $MachinePrecision] + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)} \]

   3. +-commutative N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand + 1\right)} \]

   4. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right) + 1 \cdot \left(a - \frac{1}{3}\right)} \]

   5. *-lft-identity N/A

      \[\leadsto \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right) + \color{blue}{\left(a - \frac{1}{3}\right)} \]

   6. lower-fma.f64 99.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand, a - \frac{1}{3}, a - \frac{1}{3}\right)} \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{rand}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}, a - 0.3333333333333333, a - 0.3333333333333333\right)} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -4.8 \cdot 10^{+109} \lor \neg \left(rand \leq 1.45 \cdot 10^{+77}\right):\\ \;\;\;\;\sqrt{\mathsf{fma}\left(a, 0.1111111111111111, -0.037037037037037035\right)} \cdot rand\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (or (<= rand -4.8e+109) (not (<= rand 1.45e+77)))
   (* (sqrt (fma a 0.1111111111111111 -0.037037037037037035)) rand)
   (- a 0.3333333333333333)))

C

double code(double a, double rand) {
	double tmp;
	if ((rand <= -4.8e+109) || !(rand <= 1.45e+77)) {
		tmp = sqrt(fma(a, 0.1111111111111111, -0.037037037037037035)) * rand;
	} else {
		tmp = a - 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if ((rand <= -4.8e+109) || !(rand <= 1.45e+77))
		tmp = Float64(sqrt(fma(a, 0.1111111111111111, -0.037037037037037035)) * rand);
	else
		tmp = Float64(a - 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[Or[LessEqual[rand, -4.8e+109], N[Not[LessEqual[rand, 1.45e+77]], $MachinePrecision]], N[(N[Sqrt[N[(a * 0.1111111111111111 + -0.037037037037037035), $MachinePrecision]], $MachinePrecision] * rand), $MachinePrecision], N[(a - 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < -4.79999999999999975e109 or 1.4500000000000001e77 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Add Preprocessing

   3. Taylor expanded in rand around inf

      \[\leadsto \color{blue}{rand \cdot \left(\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \frac{a}{rand}\right) - \frac{1}{3} \cdot \frac{1}{rand}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \frac{a}{rand}\right) - \frac{1}{3} \cdot \frac{1}{rand}\right) \cdot rand} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \frac{a}{rand}\right) - \frac{1}{3} \cdot \frac{1}{rand}\right) \cdot rand} \]

      3. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \left(\frac{a}{rand} - \frac{1}{3} \cdot \frac{1}{rand}\right)\right)} \cdot rand \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3}} + \left(\frac{a}{rand} - \frac{1}{3} \cdot \frac{1}{rand}\right)\right) \cdot rand \]

      5. associate-*r/ N/A

         \[\leadsto \left(\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3} + \left(\frac{a}{rand} - \color{blue}{\frac{\frac{1}{3} \cdot 1}{rand}}\right)\right) \cdot rand \]

      6. metadata-eval N/A

         \[\leadsto \left(\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3} + \left(\frac{a}{rand} - \frac{\color{blue}{\frac{1}{3}}}{rand}\right)\right) \cdot rand \]

      7. div-sub N/A

         \[\leadsto \left(\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3} + \color{blue}{\frac{a - \frac{1}{3}}{rand}}\right) \cdot rand \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{a - \frac{1}{3}}, \frac{1}{3}, \frac{a - \frac{1}{3}}{rand}\right)} \cdot rand \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{a - \frac{1}{3}}}, \frac{1}{3}, \frac{a - \frac{1}{3}}{rand}\right) \cdot rand \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{a - \frac{1}{3}}}, \frac{1}{3}, \frac{a - \frac{1}{3}}{rand}\right) \cdot rand \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{a - \frac{1}{3}}, \frac{1}{3}, \color{blue}{\frac{a - \frac{1}{3}}{rand}}\right) \cdot rand \]

      12. lower--.f64 99.5

          \[\leadsto \mathsf{fma}\left(\sqrt{a - 0.3333333333333333}, 0.3333333333333333, \frac{\color{blue}{a - 0.3333333333333333}}{rand}\right) \cdot rand \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{a - 0.3333333333333333}, 0.3333333333333333, \frac{a - 0.3333333333333333}{rand}\right) \cdot rand} \]

   6. Taylor expanded in rand around inf

      \[\leadsto \left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}}\right) \cdot rand \]
   7. Step-by-step derivation

   8. Applied rewrites 92.4%

      \[\leadsto \left(\sqrt{a - 0.3333333333333333} \cdot 0.3333333333333333\right) \cdot rand \]
   9. Step-by-step derivation

   10. Applied rewrites 92.6%

       \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \left(a - 0.3333333333333333\right)} \cdot rand} \]
   11. Step-by-step derivation

   12. Applied rewrites 92.6%

       \[\leadsto \sqrt{\mathsf{fma}\left(a, 0.1111111111111111, -0.037037037037037035\right)} \cdot rand \]

   if -4.79999999999999975e109 < rand < 1.4500000000000001e77

   1. Initial program 99.9%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Add Preprocessing

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. Step-by-step derivation

      1. lower--.f64 93.6

         \[\leadsto \color{blue}{a - 0.3333333333333333} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{a - 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -4.8 \cdot 10^{+109} \lor \neg \left(rand \leq 1.45 \cdot 10^{+77}\right):\\ \;\;\;\;\sqrt{\mathsf{fma}\left(a, 0.1111111111111111, -0.037037037037037035\right)} \cdot rand\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -4.8 \cdot 10^{+109} \lor \neg \left(rand \leq 1.45 \cdot 10^{+77}\right):\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot a} \cdot rand\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (or (<= rand -4.8e+109) (not (<= rand 1.45e+77)))
   (* (sqrt (* 0.1111111111111111 a)) rand)
   (- a 0.3333333333333333)))

C

double code(double a, double rand) {
	double tmp;
	if ((rand <= -4.8e+109) || !(rand <= 1.45e+77)) {
		tmp = sqrt((0.1111111111111111 * a)) * rand;
	} else {
		tmp = a - 0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if ((rand <= (-4.8d+109)) .or. (.not. (rand <= 1.45d+77))) then
        tmp = sqrt((0.1111111111111111d0 * a)) * rand
    else
        tmp = a - 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if ((rand <= -4.8e+109) || !(rand <= 1.45e+77)) {
		tmp = Math.sqrt((0.1111111111111111 * a)) * rand;
	} else {
		tmp = a - 0.3333333333333333;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if (rand <= -4.8e+109) or not (rand <= 1.45e+77):
		tmp = math.sqrt((0.1111111111111111 * a)) * rand
	else:
		tmp = a - 0.3333333333333333
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if ((rand <= -4.8e+109) || !(rand <= 1.45e+77))
		tmp = Float64(sqrt(Float64(0.1111111111111111 * a)) * rand);
	else
		tmp = Float64(a - 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if ((rand <= -4.8e+109) || ~((rand <= 1.45e+77)))
		tmp = sqrt((0.1111111111111111 * a)) * rand;
	else
		tmp = a - 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[Or[LessEqual[rand, -4.8e+109], N[Not[LessEqual[rand, 1.45e+77]], $MachinePrecision]], N[(N[Sqrt[N[(0.1111111111111111 * a), $MachinePrecision]], $MachinePrecision] * rand), $MachinePrecision], N[(a - 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < -4.79999999999999975e109 or 1.4500000000000001e77 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Add Preprocessing

   3. Taylor expanded in rand around inf

      \[\leadsto \color{blue}{rand \cdot \left(\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \frac{a}{rand}\right) - \frac{1}{3} \cdot \frac{1}{rand}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \frac{a}{rand}\right) - \frac{1}{3} \cdot \frac{1}{rand}\right) \cdot rand} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \frac{a}{rand}\right) - \frac{1}{3} \cdot \frac{1}{rand}\right) \cdot rand} \]

      3. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \left(\frac{a}{rand} - \frac{1}{3} \cdot \frac{1}{rand}\right)\right)} \cdot rand \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3}} + \left(\frac{a}{rand} - \frac{1}{3} \cdot \frac{1}{rand}\right)\right) \cdot rand \]

      5. associate-*r/ N/A

         \[\leadsto \left(\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3} + \left(\frac{a}{rand} - \color{blue}{\frac{\frac{1}{3} \cdot 1}{rand}}\right)\right) \cdot rand \]

      6. metadata-eval N/A

         \[\leadsto \left(\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3} + \left(\frac{a}{rand} - \frac{\color{blue}{\frac{1}{3}}}{rand}\right)\right) \cdot rand \]

      7. div-sub N/A

         \[\leadsto \left(\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3} + \color{blue}{\frac{a - \frac{1}{3}}{rand}}\right) \cdot rand \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{a - \frac{1}{3}}, \frac{1}{3}, \frac{a - \frac{1}{3}}{rand}\right)} \cdot rand \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{a - \frac{1}{3}}}, \frac{1}{3}, \frac{a - \frac{1}{3}}{rand}\right) \cdot rand \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{a - \frac{1}{3}}}, \frac{1}{3}, \frac{a - \frac{1}{3}}{rand}\right) \cdot rand \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{a - \frac{1}{3}}, \frac{1}{3}, \color{blue}{\frac{a - \frac{1}{3}}{rand}}\right) \cdot rand \]

      12. lower--.f64 99.5

          \[\leadsto \mathsf{fma}\left(\sqrt{a - 0.3333333333333333}, 0.3333333333333333, \frac{\color{blue}{a - 0.3333333333333333}}{rand}\right) \cdot rand \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{a - 0.3333333333333333}, 0.3333333333333333, \frac{a - 0.3333333333333333}{rand}\right) \cdot rand} \]

   6. Taylor expanded in rand around inf

      \[\leadsto \left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}}\right) \cdot rand \]
   7. Step-by-step derivation

   8. Applied rewrites 92.4%

      \[\leadsto \left(\sqrt{a - 0.3333333333333333} \cdot 0.3333333333333333\right) \cdot rand \]
   9. Step-by-step derivation

   10. Applied rewrites 92.6%

       \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \left(a - 0.3333333333333333\right)} \cdot rand} \]

   11. Taylor expanded in a around inf

       \[\leadsto \sqrt{\frac{1}{9} \cdot a} \cdot rand \]
   12. Step-by-step derivation

   13. Applied rewrites 89.2%

       \[\leadsto \sqrt{0.1111111111111111 \cdot a} \cdot rand \]

   if -4.79999999999999975e109 < rand < 1.4500000000000001e77

   1. Initial program 99.9%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Add Preprocessing

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. Step-by-step derivation

      1. lower--.f64 93.6

         \[\leadsto \color{blue}{a - 0.3333333333333333} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{a - 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -4.8 \cdot 10^{+109} \lor \neg \left(rand \leq 1.45 \cdot 10^{+77}\right):\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot a} \cdot rand\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -4.8 \cdot 10^{+109} \lor \neg \left(rand \leq 1.45 \cdot 10^{+77}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (or (<= rand -4.8e+109) (not (<= rand 1.45e+77)))
   (* (* (sqrt a) 0.3333333333333333) rand)
   (- a 0.3333333333333333)))

C

double code(double a, double rand) {
	double tmp;
	if ((rand <= -4.8e+109) || !(rand <= 1.45e+77)) {
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	} else {
		tmp = a - 0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if ((rand <= (-4.8d+109)) .or. (.not. (rand <= 1.45d+77))) then
        tmp = (sqrt(a) * 0.3333333333333333d0) * rand
    else
        tmp = a - 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if ((rand <= -4.8e+109) || !(rand <= 1.45e+77)) {
		tmp = (Math.sqrt(a) * 0.3333333333333333) * rand;
	} else {
		tmp = a - 0.3333333333333333;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if (rand <= -4.8e+109) or not (rand <= 1.45e+77):
		tmp = (math.sqrt(a) * 0.3333333333333333) * rand
	else:
		tmp = a - 0.3333333333333333
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if ((rand <= -4.8e+109) || !(rand <= 1.45e+77))
		tmp = Float64(Float64(sqrt(a) * 0.3333333333333333) * rand);
	else
		tmp = Float64(a - 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if ((rand <= -4.8e+109) || ~((rand <= 1.45e+77)))
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	else
		tmp = a - 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[Or[LessEqual[rand, -4.8e+109], N[Not[LessEqual[rand, 1.45e+77]], $MachinePrecision]], N[(N[(N[Sqrt[a], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision], N[(a - 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < -4.79999999999999975e109 or 1.4500000000000001e77 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Add Preprocessing

   3. Taylor expanded in rand around inf

      \[\leadsto \color{blue}{rand \cdot \left(\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \frac{a}{rand}\right) - \frac{1}{3} \cdot \frac{1}{rand}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \frac{a}{rand}\right) - \frac{1}{3} \cdot \frac{1}{rand}\right) \cdot rand} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \frac{a}{rand}\right) - \frac{1}{3} \cdot \frac{1}{rand}\right) \cdot rand} \]

      3. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \left(\frac{a}{rand} - \frac{1}{3} \cdot \frac{1}{rand}\right)\right)} \cdot rand \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3}} + \left(\frac{a}{rand} - \frac{1}{3} \cdot \frac{1}{rand}\right)\right) \cdot rand \]

      5. associate-*r/ N/A

         \[\leadsto \left(\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3} + \left(\frac{a}{rand} - \color{blue}{\frac{\frac{1}{3} \cdot 1}{rand}}\right)\right) \cdot rand \]

      6. metadata-eval N/A

         \[\leadsto \left(\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3} + \left(\frac{a}{rand} - \frac{\color{blue}{\frac{1}{3}}}{rand}\right)\right) \cdot rand \]

      7. div-sub N/A

         \[\leadsto \left(\sqrt{a - \frac{1}{3}} \cdot \frac{1}{3} + \color{blue}{\frac{a - \frac{1}{3}}{rand}}\right) \cdot rand \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{a - \frac{1}{3}}, \frac{1}{3}, \frac{a - \frac{1}{3}}{rand}\right)} \cdot rand \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{a - \frac{1}{3}}}, \frac{1}{3}, \frac{a - \frac{1}{3}}{rand}\right) \cdot rand \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{a - \frac{1}{3}}}, \frac{1}{3}, \frac{a - \frac{1}{3}}{rand}\right) \cdot rand \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{a - \frac{1}{3}}, \frac{1}{3}, \color{blue}{\frac{a - \frac{1}{3}}{rand}}\right) \cdot rand \]

      12. lower--.f64 99.5

          \[\leadsto \mathsf{fma}\left(\sqrt{a - 0.3333333333333333}, 0.3333333333333333, \frac{\color{blue}{a - 0.3333333333333333}}{rand}\right) \cdot rand \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{a - 0.3333333333333333}, 0.3333333333333333, \frac{a - 0.3333333333333333}{rand}\right) \cdot rand} \]

   6. Taylor expanded in rand around inf

      \[\leadsto \left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}}\right) \cdot rand \]
   7. Step-by-step derivation

   8. Applied rewrites 92.4%

      \[\leadsto \left(\sqrt{a - 0.3333333333333333} \cdot 0.3333333333333333\right) \cdot rand \]

   9. Taylor expanded in a around inf

      \[\leadsto \left(\frac{1}{3} \cdot \sqrt{a}\right) \cdot rand \]
   10. Step-by-step derivation

   11. Applied rewrites 89.1%

       \[\leadsto \left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand \]

   if -4.79999999999999975e109 < rand < 1.4500000000000001e77

   1. Initial program 99.9%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Add Preprocessing

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. Step-by-step derivation

      1. lower--.f64 93.6

         \[\leadsto \color{blue}{a - 0.3333333333333333} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{a - 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -4.8 \cdot 10^{+109} \lor \neg \left(rand \leq 1.45 \cdot 10^{+77}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333 \cdot rand, \sqrt{a - 0.3333333333333333}, a\right) - 0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (-
  (fma (* 0.3333333333333333 rand) (sqrt (- a 0.3333333333333333)) a)
  0.3333333333333333))

C

double code(double a, double rand) {
	return fma((0.3333333333333333 * rand), sqrt((a - 0.3333333333333333)), a) - 0.3333333333333333;
}

Julia

function code(a, rand)
	return Float64(fma(Float64(0.3333333333333333 * rand), sqrt(Float64(a - 0.3333333333333333)), a) - 0.3333333333333333)
end

Wolfram

code[a_, rand_] := N[(N[(N[(0.3333333333333333 * rand), $MachinePrecision] * N[Sqrt[N[(a - 0.3333333333333333), $MachinePrecision]], $MachinePrecision] + a), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Add Preprocessing

3. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{\left(a + \frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right)\right) - \frac{1}{3}} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(a + \frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right)\right) - \frac{1}{3}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right) + a\right)} - \frac{1}{3} \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot rand\right) \cdot \sqrt{a - \frac{1}{3}}} + a\right) - \frac{1}{3} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot rand, \sqrt{a - \frac{1}{3}}, a\right)} - \frac{1}{3} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot rand}, \sqrt{a - \frac{1}{3}}, a\right) - \frac{1}{3} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot rand, \color{blue}{\sqrt{a - \frac{1}{3}}}, a\right) - \frac{1}{3} \]

   7. lower--.f64 99.8

      \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot rand, \sqrt{\color{blue}{a - 0.3333333333333333}}, a\right) - 0.3333333333333333 \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333 \cdot rand, \sqrt{a - 0.3333333333333333}, a\right) - 0.3333333333333333} \]
6. Add Preprocessing

Alternative 7: 98.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333 \cdot rand, \sqrt{a}, a\right) - 0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (- (fma (* 0.3333333333333333 rand) (sqrt a) a) 0.3333333333333333))

C

double code(double a, double rand) {
	return fma((0.3333333333333333 * rand), sqrt(a), a) - 0.3333333333333333;
}

Julia

function code(a, rand)
	return Float64(fma(Float64(0.3333333333333333 * rand), sqrt(a), a) - 0.3333333333333333)
end

Wolfram

code[a_, rand_] := N[(N[(N[(0.3333333333333333 * rand), $MachinePrecision] * N[Sqrt[a], $MachinePrecision] + a), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Add Preprocessing

3. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{\left(a + \frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right)\right) - \frac{1}{3}} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(a + \frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right)\right) - \frac{1}{3}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right) + a\right)} - \frac{1}{3} \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot rand\right) \cdot \sqrt{a - \frac{1}{3}}} + a\right) - \frac{1}{3} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot rand, \sqrt{a - \frac{1}{3}}, a\right)} - \frac{1}{3} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot rand}, \sqrt{a - \frac{1}{3}}, a\right) - \frac{1}{3} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot rand, \color{blue}{\sqrt{a - \frac{1}{3}}}, a\right) - \frac{1}{3} \]

   7. lower--.f64 99.8

      \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot rand, \sqrt{\color{blue}{a - 0.3333333333333333}}, a\right) - 0.3333333333333333 \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333 \cdot rand, \sqrt{a - 0.3333333333333333}, a\right) - 0.3333333333333333} \]

6. Taylor expanded in a around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot rand, \sqrt{a}, a\right) - \frac{1}{3} \]
7. Step-by-step derivation

8. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot rand, \sqrt{a}, a\right) - 0.3333333333333333 \]
9. Add Preprocessing

Alternative 8: 63.0% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ a - 0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 (- a 0.3333333333333333))

C

double code(double a, double rand) {
	return a - 0.3333333333333333;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = a - 0.3333333333333333d0
end function

Java

public static double code(double a, double rand) {
	return a - 0.3333333333333333;
}

Python

def code(a, rand):
	return a - 0.3333333333333333

Julia

function code(a, rand)
	return Float64(a - 0.3333333333333333)
end

MATLAB

function tmp = code(a, rand)
	tmp = a - 0.3333333333333333;
end

Wolfram

code[a_, rand_] := N[(a - 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Add Preprocessing

3. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
4. Step-by-step derivation

   1. lower--.f64 65.7

      \[\leadsto \color{blue}{a - 0.3333333333333333} \]

5. Applied rewrites 65.7%

   \[\leadsto \color{blue}{a - 0.3333333333333333} \]
6. Add Preprocessing

Alternative 9: 1.5% accurate, 68.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 -0.3333333333333333)

C

double code(double a, double rand) {
	return -0.3333333333333333;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = -0.3333333333333333d0
end function

Java

public static double code(double a, double rand) {
	return -0.3333333333333333;
}

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

function tmp = code(a, rand)
	tmp = -0.3333333333333333;
end

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Add Preprocessing

3. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
4. Step-by-step derivation

   1. lower--.f64 65.7

      \[\leadsto \color{blue}{a - 0.3333333333333333} \]

5. Applied rewrites 65.7%

   \[\leadsto \color{blue}{a - 0.3333333333333333} \]

6. Taylor expanded in a around 0

   \[\leadsto \frac{-1}{3} \]
7. Step-by-step derivation

8. Applied rewrites 1.5%

   \[\leadsto -0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))