b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 9.1s

Alternatives: 15

Speedup: N/A×

• 41.4% 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: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision] + -1.0), $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. Taylor expanded in m around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. div-sub N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 99.9

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

5. Applied rewrites 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-*r/ N/A

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

   6. div-inv N/A

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

   7. times-frac N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. flip3-- N/A

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

   11. clear-num N/A

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

   12. clear-num N/A

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

   13. flip3-- N/A

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

   14. lift--.f64 N/A

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

   15. lower-fma.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -0.5:\\ \;\;\;\;m + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- 1.0 m) (+ -1.0 (/ (* m (- 1.0 m)) v))) -0.5)
   (+ m -1.0)
   (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (((1.0 - m) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.5) {
		tmp = m + -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) * ((-1.0d0) + ((m * (1.0d0 - m)) / v))) <= (-0.5d0)) then
        tmp = m + (-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) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.5) {
		tmp = m + -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if ((1.0 - m) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.5:
		tmp = m + -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v))) <= -0.5)
		tmp = Float64(m + -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) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.5)
		tmp = m + -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], N[(m + -1.0), $MachinePrecision], 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)) < -0.5

   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 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. lower-+.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -0.5 < (*.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. 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. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 30.0

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

   5. Applied rewrites 30.0%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 27.9%

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

   9. Taylor expanded in m around 0

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

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{m \cdot 1 + m \cdot \frac{1}{v}} \]

       2. associate-*r/ N/A

          \[\leadsto m \cdot 1 + \color{blue}{\frac{m \cdot 1}{v}} \]

       3. *-rgt-identity N/A

          \[\leadsto m \cdot 1 + \frac{\color{blue}{m}}{v} \]

       4. *-rgt-identity N/A

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

       5. lower-+.f64 N/A

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

       6. lower-/.f64 68.8

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

   11. Applied rewrites 68.8%

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

   12. Taylor expanded in v around 0

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

       1. lower-/.f64 68.8

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

   14. Applied rewrites 68.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -0.5:\\ \;\;\;\;m + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;-1 + \mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(\left(1 - m\right) \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.2e-6)
   (+ -1.0 (fma (/ m v) (fma m -2.0 1.0) m))
   (/ (* m (* (- 1.0 m) (- 1.0 m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.2e-6) {
		tmp = -1.0 + fma((m / v), fma(m, -2.0, 1.0), m);
	} else {
		tmp = (m * ((1.0 - m) * (1.0 - m))) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.2e-6)
		tmp = Float64(-1.0 + fma(Float64(m / v), fma(m, -2.0, 1.0), m));
	else
		tmp = Float64(Float64(m * Float64(Float64(1.0 - m) * Float64(1.0 - m))) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.2e-6], N[(-1.0 + N[(N[(m / v), $MachinePrecision] * N[(m * -2.0 + 1.0), $MachinePrecision] + m), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(N[(1.0 - m), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.1999999999999999e-6

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 3.1999999999999999e-6 < 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 v around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;-1 + \mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m - m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.2e-6)
   (+ -1.0 (fma (/ m v) (fma m -2.0 1.0) m))
   (* (- 1.0 m) (/ (- m (* m m)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.2e-6) {
		tmp = -1.0 + fma((m / v), fma(m, -2.0, 1.0), m);
	} else {
		tmp = (1.0 - m) * ((m - (m * m)) / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.2e-6)
		tmp = Float64(-1.0 + fma(Float64(m / v), fma(m, -2.0, 1.0), m));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m - Float64(m * m)) / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.2e-6], N[(-1.0 + N[(N[(m / v), $MachinePrecision] * N[(m * -2.0 + 1.0), $MachinePrecision] + m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.1999999999999999e-6

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 3.1999999999999999e-6 < 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

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lft-mult-inverse N/A

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

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

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. div-sub N/A

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

      12. associate-/l* N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt-out-- N/A

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

      15. *-lft-identity N/A

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

      16. unpow2 N/A

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

      17. lower--.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;-1 + \mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m - m \cdot m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   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 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. flip3-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip3-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]

   10. Applied rewrites 99.0%

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

Alternative 6: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in m around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. distribute-lft-in N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r/ N/A

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

       7. *-rgt-identity N/A

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

       8. distribute-lft1-in N/A

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

       9. +-commutative N/A

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

       10. associate-/l* N/A

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

       11. *-commutative N/A

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

       12. *-rgt-identity N/A

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

       13. times-frac N/A

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

       14. /-rgt-identity N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. +-commutative N/A

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

       18. *-commutative N/A

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

       19. lower-fma.f64 98.3

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

   11. Applied rewrites 98.3%

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

   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 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. flip3-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip3-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]

   10. Applied rewrites 99.0%

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

Alternative 7: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.42:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.42) (fma (/ m v) (fma m -2.0 1.0) -1.0) (* (/ m v) (* m m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.42) {
		tmp = fma((m / v), fma(m, -2.0, 1.0), -1.0);
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.42)
		tmp = fma(Float64(m / v), fma(m, -2.0, 1.0), -1.0);
	else
		tmp = Float64(Float64(m / v) * Float64(m * m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.42], N[(N[(m / v), $MachinePrecision] * N[(m * -2.0 + 1.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.419999999999999984

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in m around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. distribute-lft-in N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r/ N/A

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

       7. *-rgt-identity N/A

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

       8. distribute-lft1-in N/A

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

       9. +-commutative N/A

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

       10. associate-/l* N/A

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

       11. *-commutative N/A

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

       12. *-rgt-identity N/A

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

       13. times-frac N/A

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

       14. /-rgt-identity N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. +-commutative N/A

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

       18. *-commutative N/A

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

       19. lower-fma.f64 98.3

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

   11. Applied rewrites 98.3%

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

   if 0.419999999999999984 < 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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 97.2

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

   7. Applied rewrites 97.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.42:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.43) (* (- 1.0 m) (+ (/ m v) -1.0)) (* (/ m v) (* m m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} 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 <= 0.43d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    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 <= 0.43) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.43:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = (m / v) * (m * m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.43)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m / v) * Float64(m * m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.43)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = (m / v) * (m * m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.43], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.429999999999999993

   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

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

      1. lower-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 0.429999999999999993 < 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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 97.2

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

   7. Applied rewrites 97.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision] + -1.0), $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. Taylor expanded in m around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. div-sub N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 99.9

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

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

Alternative 10: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.38) (+ -1.0 (+ m (/ m v))) (* (/ m v) (* m m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = -1.0 + (m + (m / v));
	} 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 <= 0.38d0) then
        tmp = (-1.0d0) + (m + (m / v))
    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 <= 0.38) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.38:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m / v) * (m * m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.38)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m / v) * Float64(m * m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.38)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m / v) * (m * m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.38], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.38

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.38 < 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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 97.2

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

   7. Applied rewrites 97.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.35e+154) (+ -1.0 (+ m (/ 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 = -1.0 + (m + (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(-1.0 + Float64(m + Float64(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[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 77.4

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

   5. Applied rewrites 77.4%

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

   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. 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. lower-+.f64 7.3

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

   5. Applied rewrites 7.3%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 12: 76.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $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. Taylor expanded in m around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. lower-+.f64 N/A

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

   10. lower-/.f64 79.0

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

5. Applied rewrites 79.0%

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

Alternative 13: 76.2% 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. 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. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. distribute-lft-in N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*l/ N/A

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

   12. associate-/l* N/A

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

   13. distribute-rgt-out N/A

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

   14. *-rgt-identity N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 50.2

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

5. Applied rewrites 50.2%

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

6. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 50.0

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

8. Applied rewrites 50.0%

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

9. Taylor expanded in m around 0

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

    3. +-commutative N/A

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

    4. lower-+.f64 N/A

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

    5. lower-/.f64 79.0

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

11. Applied rewrites 79.0%

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

12. Final simplification 79.0%

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

Alternative 14: 27.2% 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. lower-+.f64 30.1

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

5. Applied rewrites 30.1%

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

6. Final simplification 30.1%

   \[\leadsto m + -1 \]
7. Add Preprocessing

Alternative 15: 24.8% 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 27.6%

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

Reproduce

herbie shell --seed 2024216 
(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)))