b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.8%

Time: 6.9s

Alternatives: 16

Speedup: N/A×

• 42.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

\[\begin{array}{l} \\ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * (1.0d0 - m)
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m)

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * Float64(1.0 - m))
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * (1.0d0 - m)
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m)

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * Float64(1.0 - m))
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 - m}{v} \cdot m\right) \cdot \left(1 - m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.5e-11)
   (fma (fma -2.0 m 1.0) (/ m v) -1.0)
   (* (* (/ (- 1.0 m) v) m) (- 1.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.5e-11) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), -1.0);
	} else {
		tmp = (((1.0 - m) / v) * m) * (1.0 - m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.5e-11)
		tmp = fma(fma(-2.0, m, 1.0), Float64(m / v), -1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) / v) * m) * Float64(1.0 - m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.5e-11], N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision] * m), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.5e-11

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(1 + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right)\right) - 1} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(1 \cdot m + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m\right)} - 1 \]

      2. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{m} + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m\right) - 1 \]

      3. associate--l+ N/A

         \[\leadsto \color{blue}{m + \left(\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m - 1\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m - 1\right) + m} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m - \left(1 - m\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m + \left(\mathsf{neg}\left(\left(1 - m\right)\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m + \color{blue}{-1 \cdot \left(1 - m\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{m \cdot \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right)} + -1 \cdot \left(1 - m\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(m \cdot \left(-2 \cdot \frac{m}{v}\right) + m \cdot \frac{1}{v}\right)} + -1 \cdot \left(1 - m\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(m \cdot -2\right) \cdot \frac{m}{v}} + m \cdot \frac{1}{v}\right) + -1 \cdot \left(1 - m\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(-2 \cdot m\right)} \cdot \frac{m}{v} + m \cdot \frac{1}{v}\right) + -1 \cdot \left(1 - m\right) \]

      12. associate-*r/ N/A

          \[\leadsto \left(\left(-2 \cdot m\right) \cdot \frac{m}{v} + \color{blue}{\frac{m \cdot 1}{v}}\right) + -1 \cdot \left(1 - m\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \left(\left(-2 \cdot m\right) \cdot \frac{m}{v} + \frac{\color{blue}{m}}{v}\right) + -1 \cdot \left(1 - m\right) \]

      14. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(-2 \cdot m + 1\right) \cdot \frac{m}{v}} + -1 \cdot \left(1 - m\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot m + 1, \frac{m}{v}, -1 \cdot \left(1 - m\right)\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-2, m, 1\right)}, \frac{m}{v}, -1 \cdot \left(1 - m\right)\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \color{blue}{\frac{m}{v}}, -1 \cdot \left(1 - m\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{\mathsf{neg}\left(\left(1 - m\right)\right)}\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{0 - \left(1 - m\right)}\right) \]

      20. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{\left(0 - 1\right) + m}\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{-1} + m\right) \]

      22. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{m + -1}\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m - 1\right)} \]

   6. Taylor expanded in m around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right) \]

   if 1.5e-11 < m

   1. Initial program 99.8%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} \cdot \left(1 - m\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(m \cdot \frac{1 - m}{v}\right)} \cdot \left(1 - m\right) \]

      2. div-sub N/A

         \[\leadsto \left(m \cdot \color{blue}{\left(\frac{1}{v} - \frac{m}{v}\right)}\right) \cdot \left(1 - m\right) \]

      3. unsub-neg N/A

         \[\leadsto \left(m \cdot \color{blue}{\left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{m}{v}\right)\right)\right)}\right) \cdot \left(1 - m\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(m \cdot \left(\frac{1}{v} + \color{blue}{-1 \cdot \frac{m}{v}}\right)\right) \cdot \left(1 - m\right) \]

      5. +-commutative N/A

         \[\leadsto \left(m \cdot \color{blue}{\left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)}\right) \cdot \left(1 - m\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m\right)} \cdot \left(1 - m\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m\right)} \cdot \left(1 - m\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)} \cdot m\right) \cdot \left(1 - m\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\left(\frac{1}{v} + \color{blue}{\left(\mathsf{neg}\left(\frac{m}{v}\right)\right)}\right) \cdot m\right) \cdot \left(1 - m\right) \]

      10. unsub-neg N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{v} - \frac{m}{v}\right)} \cdot m\right) \cdot \left(1 - m\right) \]

      11. div-sub N/A

          \[\leadsto \left(\color{blue}{\frac{1 - m}{v}} \cdot m\right) \cdot \left(1 - m\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1 - m}{v}} \cdot m\right) \cdot \left(1 - m\right) \]

      13. lower--.f64 99.9

          \[\leadsto \left(\frac{\color{blue}{1 - m}}{v} \cdot m\right) \cdot \left(1 - m\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\frac{1 - m}{v} \cdot m\right)} \cdot \left(1 - m\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot \left(1 - m\right) \leq -0.2:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} + m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* (- 1.0 m) m) v) 1.0) (- 1.0 m)) -0.2) -1.0 (+ (/ m v) m)))

C

double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * (1.0 - m)) <= -0.2) {
		tmp = -1.0;
	} else {
		tmp = (m / v) + m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((((((1.0d0 - m) * m) / v) - 1.0d0) * (1.0d0 - m)) <= (-0.2d0)) then
        tmp = -1.0d0
    else
        tmp = (m / v) + m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * (1.0 - m)) <= -0.2) {
		tmp = -1.0;
	} else {
		tmp = (m / v) + m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (((((1.0 - m) * m) / v) - 1.0) * (1.0 - m)) <= -0.2:
		tmp = -1.0
	else:
		tmp = (m / v) + m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(1.0 - m) * m) / v) - 1.0) * Float64(1.0 - m)) <= -0.2)
		tmp = -1.0;
	else
		tmp = Float64(Float64(m / v) + m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((((((1.0 - m) * m) / v) - 1.0) * (1.0 - m)) <= -0.2)
		tmp = -1.0;
	else
		tmp = (m / v) + m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], -0.2], -1.0, N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m)) < -0.20000000000000001

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

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

   5. Applied rewrites 97.4%

      \[\leadsto \color{blue}{-1} \]

   if -0.20000000000000001 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m))

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      4. clear-num N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      7. clear-num N/A

         \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      8. flip3-- N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      10. lower-/.f64 99.9

          \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]

      2. +-commutative N/A

         \[\leadsto m \cdot \color{blue}{\left(\frac{1}{v} + 1\right)} - 1 \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{v} \cdot m + 1 \cdot m\right)} - 1 \]

      4. associate-*l/ N/A

         \[\leadsto \left(\color{blue}{\frac{1 \cdot m}{v}} + 1 \cdot m\right) - 1 \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{\color{blue}{m}}{v} + 1 \cdot m\right) - 1 \]

      6. *-lft-identity N/A

         \[\leadsto \left(\frac{m}{v} + \color{blue}{m}\right) - 1 \]

      7. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{m}{v} + m\right)} - 1 \]

      8. lower-/.f64 67.2

         \[\leadsto \left(\color{blue}{\frac{m}{v}} + m\right) - 1 \]

   7. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\left(\frac{m}{v} + m\right) - 1} \]

   8. Taylor expanded in m around inf

      \[\leadsto m \cdot \color{blue}{\left(1 + \frac{1}{v}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.1%

       \[\leadsto \frac{m}{v} + \color{blue}{m} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot \left(1 - m\right) \leq -0.2:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} + m\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot \left(1 - m\right) \leq -0.2:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* (- 1.0 m) m) v) 1.0) (- 1.0 m)) -0.2) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * (1.0 - m)) <= -0.2) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((((((1.0d0 - m) * m) / v) - 1.0d0) * (1.0d0 - m)) <= (-0.2d0)) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * (1.0 - m)) <= -0.2) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (((((1.0 - m) * m) / v) - 1.0) * (1.0 - m)) <= -0.2:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(1.0 - m) * m) / v) - 1.0) * Float64(1.0 - m)) <= -0.2)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((((((1.0 - m) * m) / v) - 1.0) * (1.0 - m)) <= -0.2)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], -0.2], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m)) < -0.20000000000000001

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

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

   5. Applied rewrites 97.4%

      \[\leadsto \color{blue}{-1} \]

   if -0.20000000000000001 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m))

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      4. clear-num N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      7. clear-num N/A

         \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      8. flip3-- N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      10. lower-/.f64 99.9

          \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]

      2. +-commutative N/A

         \[\leadsto m \cdot \color{blue}{\left(\frac{1}{v} + 1\right)} - 1 \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{v} \cdot m + 1 \cdot m\right)} - 1 \]

      4. associate-*l/ N/A

         \[\leadsto \left(\color{blue}{\frac{1 \cdot m}{v}} + 1 \cdot m\right) - 1 \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{\color{blue}{m}}{v} + 1 \cdot m\right) - 1 \]

      6. *-lft-identity N/A

         \[\leadsto \left(\frac{m}{v} + \color{blue}{m}\right) - 1 \]

      7. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{m}{v} + m\right)} - 1 \]

      8. lower-/.f64 67.2

         \[\leadsto \left(\color{blue}{\frac{m}{v}} + m\right) - 1 \]

   7. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\left(\frac{m}{v} + m\right) - 1} \]

   8. Taylor expanded in v around 0

      \[\leadsto \frac{m}{\color{blue}{v}} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.1%

       \[\leadsto \frac{m}{\color{blue}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot \left(1 - m\right) \leq -0.2:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\frac{m}{\frac{1}{1 - m}}}{v} - 1\right) \cdot \left(1 - m\right) \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (* (- (/ (/ m (/ 1.0 (- 1.0 m))) v) 1.0) (- 1.0 m)))

C

double code(double m, double v) {
	return (((m / (1.0 / (1.0 - m))) / v) - 1.0) * (1.0 - m);
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m / (1.0d0 / (1.0d0 - m))) / v) - 1.0d0) * (1.0d0 - m)
end function

Java

public static double code(double m, double v) {
	return (((m / (1.0 / (1.0 - m))) / v) - 1.0) * (1.0 - m);
}

Python

def code(m, v):
	return (((m / (1.0 / (1.0 - m))) / v) - 1.0) * (1.0 - m)

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m / Float64(1.0 / Float64(1.0 - m))) / v) - 1.0) * Float64(1.0 - m))
end

MATLAB

function tmp = code(m, v)
	tmp = (((m / (1.0 / (1.0 - m))) / v) - 1.0) * (1.0 - m);
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m / N[(1.0 / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

   2. lift--.f64 N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

   3. flip3-- N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. clear-num N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. un-div-inv N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   7. clear-num N/A

      \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   8. flip3-- N/A

      \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   9. lift--.f64 N/A

      \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   10. lower-/.f64 99.9

       \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.75 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot \left(1 - m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.75e-8)
   (fma (fma -2.0 m 1.0) (/ m v) -1.0)
   (/ (* (* (- 1.0 m) m) (- 1.0 m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.75e-8) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), -1.0);
	} else {
		tmp = (((1.0 - m) * m) * (1.0 - m)) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.75e-8)
		tmp = fma(fma(-2.0, m, 1.0), Float64(m / v), -1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) * m) * Float64(1.0 - m)) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.75e-8], N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.7500000000000001e-8

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(1 + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right)\right) - 1} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(1 \cdot m + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m\right)} - 1 \]

      2. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{m} + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m\right) - 1 \]

      3. associate--l+ N/A

         \[\leadsto \color{blue}{m + \left(\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m - 1\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m - 1\right) + m} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m - \left(1 - m\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m + \left(\mathsf{neg}\left(\left(1 - m\right)\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m + \color{blue}{-1 \cdot \left(1 - m\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{m \cdot \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right)} + -1 \cdot \left(1 - m\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(m \cdot \left(-2 \cdot \frac{m}{v}\right) + m \cdot \frac{1}{v}\right)} + -1 \cdot \left(1 - m\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(m \cdot -2\right) \cdot \frac{m}{v}} + m \cdot \frac{1}{v}\right) + -1 \cdot \left(1 - m\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(-2 \cdot m\right)} \cdot \frac{m}{v} + m \cdot \frac{1}{v}\right) + -1 \cdot \left(1 - m\right) \]

      12. associate-*r/ N/A

          \[\leadsto \left(\left(-2 \cdot m\right) \cdot \frac{m}{v} + \color{blue}{\frac{m \cdot 1}{v}}\right) + -1 \cdot \left(1 - m\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \left(\left(-2 \cdot m\right) \cdot \frac{m}{v} + \frac{\color{blue}{m}}{v}\right) + -1 \cdot \left(1 - m\right) \]

      14. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(-2 \cdot m + 1\right) \cdot \frac{m}{v}} + -1 \cdot \left(1 - m\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot m + 1, \frac{m}{v}, -1 \cdot \left(1 - m\right)\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-2, m, 1\right)}, \frac{m}{v}, -1 \cdot \left(1 - m\right)\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \color{blue}{\frac{m}{v}}, -1 \cdot \left(1 - m\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{\mathsf{neg}\left(\left(1 - m\right)\right)}\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{0 - \left(1 - m\right)}\right) \]

      20. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{\left(0 - 1\right) + m}\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{-1} + m\right) \]

      22. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{m + -1}\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m - 1\right)} \]

   6. Taylor expanded in m around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right) \]

   if 2.7500000000000001e-8 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      4. clear-num N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      7. clear-num N/A

         \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      8. flip3-- N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      10. lower-/.f64 99.9

          \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\left(1 - m\right) \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)}{v}} \]

   8. Taylor expanded in v around 0

      \[\leadsto \frac{\left(1 - m\right) \cdot \left(m - {m}^{2}\right)}{v} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \frac{\left(1 - m\right) \cdot \left(\left(1 - m\right) \cdot m\right)}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.75 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot \left(1 - m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.41:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} \cdot m\right) \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.41) (fma (fma -2.0 m 1.0) (/ m v) -1.0) (* (* (/ m v) m) m)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.41) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), -1.0);
	} else {
		tmp = ((m / v) * m) * m;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.41)
		tmp = fma(fma(-2.0, m, 1.0), Float64(m / v), -1.0);
	else
		tmp = Float64(Float64(Float64(m / v) * m) * m);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.41], N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.409999999999999976

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(1 + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right)\right) - 1} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(1 \cdot m + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m\right)} - 1 \]

      2. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{m} + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m\right) - 1 \]

      3. associate--l+ N/A

         \[\leadsto \color{blue}{m + \left(\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m - 1\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m - 1\right) + m} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m - \left(1 - m\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m + \left(\mathsf{neg}\left(\left(1 - m\right)\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right) \cdot m + \color{blue}{-1 \cdot \left(1 - m\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{m \cdot \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right)} + -1 \cdot \left(1 - m\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(m \cdot \left(-2 \cdot \frac{m}{v}\right) + m \cdot \frac{1}{v}\right)} + -1 \cdot \left(1 - m\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(m \cdot -2\right) \cdot \frac{m}{v}} + m \cdot \frac{1}{v}\right) + -1 \cdot \left(1 - m\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(-2 \cdot m\right)} \cdot \frac{m}{v} + m \cdot \frac{1}{v}\right) + -1 \cdot \left(1 - m\right) \]

      12. associate-*r/ N/A

          \[\leadsto \left(\left(-2 \cdot m\right) \cdot \frac{m}{v} + \color{blue}{\frac{m \cdot 1}{v}}\right) + -1 \cdot \left(1 - m\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \left(\left(-2 \cdot m\right) \cdot \frac{m}{v} + \frac{\color{blue}{m}}{v}\right) + -1 \cdot \left(1 - m\right) \]

      14. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(-2 \cdot m + 1\right) \cdot \frac{m}{v}} + -1 \cdot \left(1 - m\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot m + 1, \frac{m}{v}, -1 \cdot \left(1 - m\right)\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-2, m, 1\right)}, \frac{m}{v}, -1 \cdot \left(1 - m\right)\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \color{blue}{\frac{m}{v}}, -1 \cdot \left(1 - m\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{\mathsf{neg}\left(\left(1 - m\right)\right)}\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{0 - \left(1 - m\right)}\right) \]

      20. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{\left(0 - 1\right) + m}\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{-1} + m\right) \]

      22. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{m + -1}\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m - 1\right)} \]

   6. Taylor expanded in m around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right) \]

   if 0.409999999999999976 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \frac{\color{blue}{\left(m \cdot m\right) \cdot m}}{v} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{m}^{2}} \cdot m}{v} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v} \cdot m} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v} \cdot m} \]

      5. unpow2 N/A

         \[\leadsto \frac{\color{blue}{m \cdot m}}{v} \cdot m \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{m}{v} \cdot m\right)} \cdot m \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{m}{v} \cdot m\right)} \cdot m \]

      8. lower-/.f64 96.4

         \[\leadsto \left(\color{blue}{\frac{m}{v}} \cdot m\right) \cdot m \]

   5. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left(\frac{m}{v} \cdot m\right) \cdot m} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} \cdot m\right) \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0) (* (- (/ m v) 1.0) (- 1.0 m)) (* (* (/ m v) m) m)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((m / v) * m) * m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.0d0) then
        tmp = ((m / v) - 1.0d0) * (1.0d0 - m)
    else
        tmp = ((m / v) * m) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((m / v) * m) * m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = ((m / v) - 1.0) * (1.0 - m)
	else:
		tmp = ((m / v) * m) * m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * Float64(1.0 - m));
	else
		tmp = Float64(Float64(Float64(m / v) * m) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = ((m / v) - 1.0) * (1.0 - m);
	else
		tmp = ((m / v) * m) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot \left(1 - m\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 97.5

         \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot \left(1 - m\right) \]

   5. Applied rewrites 97.5%

      \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot \left(1 - m\right) \]

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \frac{\color{blue}{\left(m \cdot m\right) \cdot m}}{v} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{m}^{2}} \cdot m}{v} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v} \cdot m} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v} \cdot m} \]

      5. unpow2 N/A

         \[\leadsto \frac{\color{blue}{m \cdot m}}{v} \cdot m \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{m}{v} \cdot m\right)} \cdot m \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{m}{v} \cdot m\right)} \cdot m \]

      8. lower-/.f64 97.0

         \[\leadsto \left(\color{blue}{\frac{m}{v}} \cdot m\right) \cdot m \]

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\left(\frac{m}{v} \cdot m\right) \cdot m} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(m - \mathsf{fma}\left(m, m, v\right)\right) \cdot \left(1 - m\right)}{v} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (/ (* (- m (fma m m v)) (- 1.0 m)) v))

C

double code(double m, double v) {
	return ((m - fma(m, m, v)) * (1.0 - m)) / v;
}

Julia

function code(m, v)
	return Float64(Float64(Float64(m - fma(m, m, v)) * Float64(1.0 - m)) / v)
end

Wolfram

code[m_, v_] := N[(N[(N[(m - N[(m * m + v), $MachinePrecision]), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

   2. lift--.f64 N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

   3. flip3-- N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. clear-num N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. un-div-inv N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   7. clear-num N/A

      \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   8. flip3-- N/A

      \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   9. lift--.f64 N/A

      \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   10. lower-/.f64 99.9

       \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]

7. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\left(1 - m\right) \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)}{v}} \]

8. Final simplification 99.9%

   \[\leadsto \frac{\left(m - \mathsf{fma}\left(m, m, v\right)\right) \cdot \left(1 - m\right)}{v} \]
9. Add Preprocessing

Alternative 9: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, v, m\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} \cdot m\right) \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6) (/ (fma -1.0 v m) v) (* (* (/ m v) m) m)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = fma(-1.0, v, m) / v;
	} else {
		tmp = ((m / v) * m) * m;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6)
		tmp = Float64(fma(-1.0, v, m) / v);
	else
		tmp = Float64(Float64(Float64(m / v) * m) * m);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6], N[(N[(-1.0 * v + m), $MachinePrecision] / v), $MachinePrecision], N[(N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000009

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      4. clear-num N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      7. clear-num N/A

         \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      8. flip3-- N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      10. lower-/.f64 99.9

          \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\left(1 - m\right) \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)}{v}} \]

   8. Taylor expanded in m around 0

      \[\leadsto \frac{-1 \cdot v + m \cdot \left(1 + v\right)}{v} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.8%

       \[\leadsto \frac{\mathsf{fma}\left(m - 1, v, m\right)}{v} \]

   11. Taylor expanded in m around 0

       \[\leadsto \frac{\mathsf{fma}\left(-1, v, m\right)}{v} \]
   12. Step-by-step derivation

   13. Applied rewrites 96.8%

       \[\leadsto \frac{\mathsf{fma}\left(-1, v, m\right)}{v} \]

   if 2.60000000000000009 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \frac{\color{blue}{\left(m \cdot m\right) \cdot m}}{v} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{m}^{2}} \cdot m}{v} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v} \cdot m} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v} \cdot m} \]

      5. unpow2 N/A

         \[\leadsto \frac{\color{blue}{m \cdot m}}{v} \cdot m \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{m}{v} \cdot m\right)} \cdot m \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{m}{v} \cdot m\right)} \cdot m \]

      8. lower-/.f64 97.7

         \[\leadsto \left(\color{blue}{\frac{m}{v}} \cdot m\right) \cdot m \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\left(\frac{m}{v} \cdot m\right) \cdot m} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m - 1, v, m\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m, m, -1\right)}{m - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.35e+154) (/ (fma (- m 1.0) v m) v) (/ (fma m m -1.0) (- m -1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.35e+154) {
		tmp = fma((m - 1.0), v, m) / v;
	} else {
		tmp = fma(m, m, -1.0) / (m - -1.0);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.35e+154)
		tmp = Float64(fma(Float64(m - 1.0), v, m) / v);
	else
		tmp = Float64(fma(m, m, -1.0) / Float64(m - -1.0));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.35e+154], N[(N[(N[(m - 1.0), $MachinePrecision] * v + m), $MachinePrecision] / v), $MachinePrecision], N[(N[(m * m + -1.0), $MachinePrecision] / N[(m - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.35000000000000003e154

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      4. clear-num N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      7. clear-num N/A

         \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      8. flip3-- N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      10. lower-/.f64 99.9

          \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\left(1 - m\right) \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)}{v}} \]

   8. Taylor expanded in m around 0

      \[\leadsto \frac{-1 \cdot v + m \cdot \left(1 + v\right)}{v} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.0%

       \[\leadsto \frac{\mathsf{fma}\left(m - 1, v, m\right)}{v} \]

   if 1.35000000000000003e154 < m

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      4. clear-num N/A

         \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      7. clear-num N/A

         \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      8. flip3-- N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

      10. lower-/.f64 100.0

          \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(1 - m\right)\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(1 - m\right)} \]

      3. associate--r- N/A

         \[\leadsto \color{blue}{\left(0 - 1\right) + m} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{-1} + m \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{m + -1} \]

      6. metadata-eval N/A

         \[\leadsto m + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \color{blue}{m - 1} \]

      8. lower--.f64 7.2

         \[\leadsto \color{blue}{m - 1} \]

   7. Applied rewrites 7.2%

      \[\leadsto \color{blue}{m - 1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(m, m, -1\right)}{\color{blue}{m - -1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(m - 1, v, m\right)}{v} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (/ (fma (- m 1.0) v m) v))

C

double code(double m, double v) {
	return fma((m - 1.0), v, m) / v;
}

Julia

function code(m, v)
	return Float64(fma(Float64(m - 1.0), v, m) / v)
end

Wolfram

code[m_, v_] := N[(N[(N[(m - 1.0), $MachinePrecision] * v + m), $MachinePrecision] / v), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

   2. lift--.f64 N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

   3. flip3-- N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. clear-num N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. un-div-inv N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   7. clear-num N/A

      \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   8. flip3-- N/A

      \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   9. lift--.f64 N/A

      \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   10. lower-/.f64 99.9

       \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]

7. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\left(1 - m\right) \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)}{v}} \]

8. Taylor expanded in m around 0

   \[\leadsto \frac{-1 \cdot v + m \cdot \left(1 + v\right)}{v} \]
9. Step-by-step derivation

10. Applied rewrites 74.9%

    \[\leadsto \frac{\mathsf{fma}\left(m - 1, v, m\right)}{v} \]
11. Add Preprocessing

Alternative 12: 75.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-1, v, m\right)}{v} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (/ (fma -1.0 v m) v))

C

double code(double m, double v) {
	return fma(-1.0, v, m) / v;
}

Julia

function code(m, v)
	return Float64(fma(-1.0, v, m) / v)
end

Wolfram

code[m_, v_] := N[(N[(-1.0 * v + m), $MachinePrecision] / v), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

   2. lift--.f64 N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

   3. flip3-- N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. clear-num N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. un-div-inv N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   7. clear-num N/A

      \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   8. flip3-- N/A

      \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   9. lift--.f64 N/A

      \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   10. lower-/.f64 99.9

       \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(v \cdot \left(1 - m\right)\right) + m \cdot {\left(1 - m\right)}^{2}}{v}} \]

7. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\left(1 - m\right) \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)}{v}} \]

8. Taylor expanded in m around 0

   \[\leadsto \frac{-1 \cdot v + m \cdot \left(1 + v\right)}{v} \]
9. Step-by-step derivation

10. Applied rewrites 74.9%

    \[\leadsto \frac{\mathsf{fma}\left(m - 1, v, m\right)}{v} \]

11. Taylor expanded in m around 0

    \[\leadsto \frac{\mathsf{fma}\left(-1, v, m\right)}{v} \]
12. Step-by-step derivation

13. Applied rewrites 74.9%

    \[\leadsto \frac{\mathsf{fma}\left(-1, v, m\right)}{v} \]
14. Add Preprocessing

Alternative 13: 75.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{m}{v} + m\right) - 1 \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (- (+ (/ m v) m) 1.0))

C

double code(double m, double v) {
	return ((m / v) + m) - 1.0;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = ((m / v) + m) - 1.0d0
end function

Java

public static double code(double m, double v) {
	return ((m / v) + m) - 1.0;
}

Python

def code(m, v):
	return ((m / v) + m) - 1.0

Julia

function code(m, v)
	return Float64(Float64(Float64(m / v) + m) - 1.0)
end

MATLAB

function tmp = code(m, v)
	tmp = ((m / v) + m) - 1.0;
end

Wolfram

code[m_, v_] := N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Add Preprocessing

3. Taylor expanded in m around 0

   \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]

   2. +-commutative N/A

      \[\leadsto m \cdot \color{blue}{\left(\frac{1}{v} + 1\right)} - 1 \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{v} \cdot m + 1 \cdot m\right)} - 1 \]

   4. associate-*l/ N/A

      \[\leadsto \left(\color{blue}{\frac{1 \cdot m}{v}} + 1 \cdot m\right) - 1 \]

   5. *-lft-identity N/A

      \[\leadsto \left(\frac{\color{blue}{m}}{v} + 1 \cdot m\right) - 1 \]

   6. *-lft-identity N/A

      \[\leadsto \left(\frac{m}{v} + \color{blue}{m}\right) - 1 \]

   7. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{m}{v} + m\right)} - 1 \]

   8. lower-/.f64 74.9

      \[\leadsto \left(\color{blue}{\frac{m}{v}} + m\right) - 1 \]

5. Applied rewrites 74.9%

   \[\leadsto \color{blue}{\left(\frac{m}{v} + m\right) - 1} \]
6. Add Preprocessing

Alternative 14: 75.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{m}{v} - 1 \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (- (/ m v) 1.0))

C

double code(double m, double v) {
	return (m / v) - 1.0;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (m / v) - 1.0d0
end function

Java

public static double code(double m, double v) {
	return (m / v) - 1.0;
}

Python

def code(m, v):
	return (m / v) - 1.0

Julia

function code(m, v)
	return Float64(Float64(m / v) - 1.0)
end

MATLAB

function tmp = code(m, v)
	tmp = (m / v) - 1.0;
end

Wolfram

code[m_, v_] := N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

   2. lift--.f64 N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 - m\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]

   3. flip3-- N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. clear-num N/A

      \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   5. un-div-inv N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}{{1}^{3} - {m}^{3}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   7. clear-num N/A

      \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{\frac{{1}^{3} - {m}^{3}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)}}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   8. flip3-- N/A

      \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   9. lift--.f64 N/A

      \[\leadsto \left(\frac{\frac{m}{\frac{1}{\color{blue}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

   10. lower-/.f64 99.9

       \[\leadsto \left(\frac{\frac{m}{\color{blue}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \left(\frac{\color{blue}{\frac{m}{\frac{1}{1 - m}}}}{v} - 1\right) \cdot \left(1 - m\right) \]

5. Taylor expanded in m around 0

   \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]

   2. +-commutative N/A

      \[\leadsto m \cdot \color{blue}{\left(\frac{1}{v} + 1\right)} - 1 \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{v} \cdot m + 1 \cdot m\right)} - 1 \]

   4. associate-*l/ N/A

      \[\leadsto \left(\color{blue}{\frac{1 \cdot m}{v}} + 1 \cdot m\right) - 1 \]

   5. *-lft-identity N/A

      \[\leadsto \left(\frac{\color{blue}{m}}{v} + 1 \cdot m\right) - 1 \]

   6. *-lft-identity N/A

      \[\leadsto \left(\frac{m}{v} + \color{blue}{m}\right) - 1 \]

   7. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{m}{v} + m\right)} - 1 \]

   8. lower-/.f64 74.9

      \[\leadsto \left(\color{blue}{\frac{m}{v}} + m\right) - 1 \]

7. Applied rewrites 74.9%

   \[\leadsto \color{blue}{\left(\frac{m}{v} + m\right) - 1} \]

8. Taylor expanded in v around 0

   \[\leadsto \frac{m}{v} - 1 \]
9. Step-by-step derivation

10. Applied rewrites 74.9%

    \[\leadsto \frac{m}{v} - 1 \]
11. Add Preprocessing

Alternative 15: 27.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ m - 1 \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (- m 1.0))

C

double code(double m, double v) {
	return m - 1.0;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = m - 1.0d0
end function

Java

public static double code(double m, double v) {
	return m - 1.0;
}

Python

def code(m, v):
	return m - 1.0

Julia

function code(m, v)
	return Float64(m - 1.0)
end

MATLAB

function tmp = code(m, v)
	tmp = m - 1.0;
end

Wolfram

code[m_, v_] := N[(m - 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Add Preprocessing

3. Taylor expanded in v around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(1 - m\right)\right)} \]

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \left(1 - m\right)} \]

   3. associate--r- N/A

      \[\leadsto \color{blue}{\left(0 - 1\right) + m} \]

   4. metadata-eval N/A

      \[\leadsto \color{blue}{-1} + m \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{m + -1} \]

   6. metadata-eval N/A

      \[\leadsto m + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \color{blue}{m - 1} \]

   8. lower--.f64 25.7

      \[\leadsto \color{blue}{m - 1} \]

5. Applied rewrites 25.7%

   \[\leadsto \color{blue}{m - 1} \]
6. Add Preprocessing

Alternative 16: 25.2% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (m v) :precision binary64 -1.0)

C

double code(double m, double v) {
	return -1.0;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = -1.0d0
end function

Java

public static double code(double m, double v) {
	return -1.0;
}

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

function tmp = code(m, v)
	tmp = -1.0;
end

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Add Preprocessing

3. Taylor expanded in m around 0

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

5. Applied rewrites 23.5%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))