b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

Alternatives: 16

Speedup: N/A×

• 42.4% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\left(1 - m\right)}^{2}, \frac{m}{v}, \mathsf{fma}\left(-1, m, -1\right)\right) \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (fma (pow (- 1.0 m) 2.0) (/ m v) (fma -1.0 m -1.0)))

C

double code(double m, double v) {
	return fma(pow((1.0 - m), 2.0), (m / v), fma(-1.0, m, -1.0));
}

Julia

function code(m, v)
	return fma((Float64(1.0 - m) ^ 2.0), Float64(m / v), fma(-1.0, m, -1.0))
end

Wolfram

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

4. Applied rewrites 100.0%

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

Alternative 2: 73.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((((((1.0d0 - m) * m) / v) - 1.0d0) * (1.0d0 - m)) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = (m / v) + m
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m)) < -0.5

   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 93.7%

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

   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 m around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 66.2%

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

4. Final simplification 73.5%

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

Alternative 3: 73.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((((((1.0d0 - m) * m) / v) - 1.0d0) * (1.0d0 - m)) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m)) < -0.5

   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 93.7%

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

   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 m around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 66.2%

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

4. Final simplification 73.5%

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

Alternative 4: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, v\right), m, m - v\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot \left(1 - m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-9)
   (/ (fma (fma -2.0 m v) m (- m v)) v)
   (/ (* (* (- 1.0 m) m) (- 1.0 m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-9) {
		tmp = fma(fma(-2.0, m, v), m, (m - v)) / v;
	} else {
		tmp = (((1.0 - m) * m) * (1.0 - m)) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-9)
		tmp = Float64(fma(fma(-2.0, m, v), m, Float64(m - v)) / v);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) * m) * Float64(1.0 - m)) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-9], N[(N[(N[(-2.0 * m + v), $MachinePrecision] * m + N[(m - v), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.0000000000000001e-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

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 100.0%

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

   if 5.0000000000000001e-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

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, v\right), m, m - v\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot \left(1 - m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, v\right), m, m - v\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(-m\right) \cdot \left(\frac{m}{v} \cdot \left(1 - m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.62)
   (/ (fma (fma -2.0 m v) m (- m v)) v)
   (* (- m) (* (/ m v) (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.62) {
		tmp = fma(fma(-2.0, m, v), m, (m - v)) / v;
	} else {
		tmp = -m * ((m / v) * (1.0 - m));
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.62)
		tmp = Float64(fma(fma(-2.0, m, v), m, Float64(m - v)) / v);
	else
		tmp = Float64(Float64(-m) * Float64(Float64(m / v) * Float64(1.0 - m)));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.62], N[(N[(N[(-2.0 * m + v), $MachinePrecision] * m + N[(m - v), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision], N[((-m) * N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.619999999999999996

   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 51.1%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 98.6%

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

   if 0.619999999999999996 < 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. associate-/l* N/A

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

      2. div-sub N/A

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

      3. unsub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. div-sub N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in m around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.0

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

   8. Applied rewrites 98.0%

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

   10. Applied rewrites 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, v\right), m, m - v\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(-m\right) \cdot \left(\frac{m}{v} \cdot \left(1 - m\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, v\right), m, m - v\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-m\right) \cdot m\right) \cdot \left(1 - m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.62)
   (/ (fma (fma -2.0 m v) m (- m v)) v)
   (/ (* (* (- m) m) (- 1.0 m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.62) {
		tmp = fma(fma(-2.0, m, v), m, (m - v)) / v;
	} else {
		tmp = ((-m * m) * (1.0 - m)) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.62)
		tmp = Float64(fma(fma(-2.0, m, v), m, Float64(m - v)) / v);
	else
		tmp = Float64(Float64(Float64(Float64(-m) * m) * Float64(1.0 - m)) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.62], N[(N[(N[(-2.0 * m + v), $MachinePrecision] * m + N[(m - v), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision], N[(N[(N[((-m) * m), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.619999999999999996

   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 51.1%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 98.6%

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

   if 0.619999999999999996 < 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

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, v\right), m, m - v\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-m\right) \cdot m\right) \cdot \left(1 - m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, v\right), m, m - v\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{v} \cdot \left(\left(-m\right) \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.62)
   (/ (fma (fma -2.0 m v) m (- m v)) v)
   (* (/ (- 1.0 m) v) (* (- m) m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.62) {
		tmp = fma(fma(-2.0, m, v), m, (m - v)) / v;
	} else {
		tmp = ((1.0 - m) / v) * (-m * m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.62)
		tmp = Float64(fma(fma(-2.0, m, v), m, Float64(m - v)) / v);
	else
		tmp = Float64(Float64(Float64(1.0 - m) / v) * Float64(Float64(-m) * m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.62], N[(N[(N[(-2.0 * m + v), $MachinePrecision] * m + N[(m - v), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision] * N[((-m) * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.619999999999999996

   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 51.1%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 98.6%

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

   if 0.619999999999999996 < 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

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 98.0%

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

   13. Applied rewrites 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, m, v\right), m, m - v\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{v} \cdot \left(\left(-m\right) \cdot m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{v} \cdot \left(\left(-m\right) \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.62)
   (fma (fma -2.0 m 1.0) (/ m v) -1.0)
   (* (/ (- 1.0 m) v) (* (- m) m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.62) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), -1.0);
	} else {
		tmp = ((1.0 - m) / v) * (-m * m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.62)
		tmp = fma(fma(-2.0, m, 1.0), Float64(m / v), -1.0);
	else
		tmp = Float64(Float64(Float64(1.0 - m) / v) * Float64(Float64(-m) * m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.62], N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision] * N[((-m) * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.619999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. mul-1-neg N/A

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

      19. neg-sub0 N/A

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

      20. associate--r- N/A

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

      21. metadata-eval N/A

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

      22. +-commutative N/A

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

      23. metadata-eval N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 98.6%

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

   if 0.619999999999999996 < 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

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 98.0%

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

   13. Applied rewrites 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{v} \cdot \left(\left(-m\right) \cdot m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.419999999999999984

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. mul-1-neg N/A

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

      19. neg-sub0 N/A

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

      20. associate--r- N/A

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

      21. metadata-eval N/A

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

      22. +-commutative N/A

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

      23. metadata-eval N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 98.6%

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

   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 v around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. unsub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. div-sub N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in m around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.0

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 98.0%

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

Alternative 10: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.419999999999999984

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. mul-1-neg N/A

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

      19. neg-sub0 N/A

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

      20. associate--r- N/A

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

      21. metadata-eval N/A

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

      22. +-commutative N/A

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

      23. metadata-eval N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 98.6%

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

   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 0

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 98.0%

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

   12. Taylor expanded in m around inf

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

   14. Applied rewrites 98.0%

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

4. Final simplification 98.3%

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

Alternative 11: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. distribute-lft-out-- N/A

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

   6. *-rgt-identity N/A

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

   7. unpow2 N/A

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

   8. associate--l- N/A

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

   9. lower--.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 12: 80.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.12 \cdot 10^{+148}:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m, m, -1\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.12e+148) (- (+ (/ m v) m) 1.0) (/ (fma m m -1.0) 1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.12e+148) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = fma(m, m, -1.0) / 1.0;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.12e+148)
		tmp = Float64(Float64(Float64(m / v) + m) - 1.0);
	else
		tmp = Float64(fma(m, m, -1.0) / 1.0);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.12e+148], N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(m * m + -1.0), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.12e148

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

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

   if 1.12e148 < 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. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 6.7

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

   5. Applied rewrites 6.7%

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 93.9%

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

Alternative 13: 74.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 76.0

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

5. Applied rewrites 76.0%

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

Alternative 14: 74.9% 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

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 76.0

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

5. Applied rewrites 76.0%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 76.0%

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

Alternative 15: 26.9% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in v around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. associate--r- N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 28.0

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

5. Applied rewrites 28.0%

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

Alternative 16: 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 25.5%

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

Reproduce

herbie shell --seed 2024253 
(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)))