b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 4.7s

Alternatives: 10

Speedup: 1.0×

• 41.0% 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: 99.9% 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]

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. Add Preprocessing

Alternative 2: 73.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(N[(m / v), $MachinePrecision] * 1.0), $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 m around 0

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

   5. Applied rewrites 92.5%

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

   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 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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 35.0%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 66.4%

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

4. Final simplification 72.8%

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

Alternative 3: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 - m}{v} \cdot m\right) \cdot \left(1 - m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 9.5e-9)
   (- (fma (fma -2.0 m 1.0) (/ m v) m) 1.0)
   (* (* (/ (- 1.0 m) v) m) (- 1.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 9.5e-9) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), m) - 1.0;
	} else {
		tmp = (((1.0 - m) / v) * m) * (1.0 - m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 9.5e-9)
		tmp = Float64(fma(fma(-2.0, m, 1.0), Float64(m / v), m) - 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, 9.5e-9], N[(N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + m), $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 < 9.5000000000000007e-9

   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. 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. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 9.5000000000000007e-9 < 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

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m - 2, m, 1\right) \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 9.5e-9)
   (- (fma (fma -2.0 m 1.0) (/ m v) m) 1.0)
   (* (fma (- m 2.0) m 1.0) (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 9.5e-9) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), m) - 1.0;
	} else {
		tmp = fma((m - 2.0), m, 1.0) * (m / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 9.5e-9)
		tmp = Float64(fma(fma(-2.0, m, 1.0), Float64(m / v), m) - 1.0);
	else
		tmp = Float64(fma(Float64(m - 2.0), m, 1.0) * Float64(m / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 9.5e-9], N[(N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(m - 2.0), $MachinePrecision] * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.5000000000000007e-9

   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. 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. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 9.5000000000000007e-9 < 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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 99.9%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.46 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m - 2, m, 1\right) \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.46e-15)
   (* (- (/ m v) 1.0) (- 1.0 m))
   (* (fma (- m 2.0) m 1.0) (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.46e-15) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = fma((m - 2.0), m, 1.0) * (m / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.46e-15)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * Float64(1.0 - m));
	else
		tmp = Float64(fma(Float64(m - 2.0), m, 1.0) * Float64(m / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.46e-15], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m - 2.0), $MachinePrecision] * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.4600000000000001e-15

   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.4600000000000001e-15 < 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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 99.9%

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

Alternative 6: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m - 1}{m - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.34e+154) (- (+ (/ m v) m) 1.0) (/ (- (* m m) 1.0) (- m -1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.34e+154) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = ((m * m) - 1.0) / (m - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.34d+154) then
        tmp = ((m / v) + m) - 1.0d0
    else
        tmp = ((m * m) - 1.0d0) / (m - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.34e+154) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = ((m * m) - 1.0) / (m - -1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.34e+154:
		tmp = ((m / v) + m) - 1.0
	else:
		tmp = ((m * m) - 1.0) / (m - -1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.34e+154)
		tmp = Float64(Float64(Float64(m / v) + m) - 1.0);
	else
		tmp = Float64(Float64(Float64(m * m) - 1.0) / Float64(m - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.34e+154)
		tmp = ((m / v) + m) - 1.0;
	else
		tmp = ((m * m) - 1.0) / (m - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.34e+154], N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(m * m), $MachinePrecision] - 1.0), $MachinePrecision] / N[(m - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.34000000000000001e154

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if 1.34000000000000001e154 < 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. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-+.f64 6.9

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

   5. Applied rewrites 6.9%

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

   7. Applied rewrites 100.0%

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

Alternative 7: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. *-inverses N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-/.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 8: 75.5% 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. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. lower--.f64 N/A

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 76.2

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

5. Applied rewrites 76.2%

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

Alternative 9: 26.8% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(-1.0 + m), $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. *-lft-identity N/A

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

   3. metadata-eval N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. distribute-neg-in N/A

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

   6. metadata-eval N/A

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

   7. mul-1-neg N/A

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

   8. remove-double-neg N/A

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

   9. lower-+.f64 25.7

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

5. Applied rewrites 25.7%

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

Alternative 10: 24.4% 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 2024339 
(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)))