b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

Alternatives: 12

Speedup: 1.0×

• 40.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-19}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2e-19) (+ -1.0 (/ m v)) (/ (* (- 1.0 m) (- m (* m m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2e-19) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = ((1.0 - m) * (m - (m * 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 (m <= 2d-19) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = ((1.0d0 - m) * (m - (m * m))) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2e-19) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = ((1.0 - m) * (m - (m * m))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2e-19:
		tmp = -1.0 + (m / v)
	else:
		tmp = ((1.0 - m) * (m - (m * m))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2e-19)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(m - Float64(m * m))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2e-19)
		tmp = -1.0 + (m / v);
	else
		tmp = ((1.0 - m) * (m - (m * m))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2e-19], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2e-19

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. div-inv 99.9%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. *-lft-identity 99.9%

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

      5. associate-*l/ 100.0%

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

      6. *-lft-identity 100.0%

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

      7. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 100.0%

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

   if 2e-19 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/r/ 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-rgt-in 54.1%

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

      4. *-un-lft-identity 54.1%

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

      5. div-inv 54.1%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in v around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. pow2 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-19}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-20}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m + \left(m \cdot m\right) \cdot \left(m + -2\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-20) (+ -1.0 (/ m v)) (/ (+ m (* (* m m) (+ m -2.0))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-20) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m + ((m * m) * (m + -2.0))) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 5d-20) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = (m + ((m * m) * (m + (-2.0d0)))) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5e-20) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m + ((m * m) * (m + -2.0))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5e-20:
		tmp = -1.0 + (m / v)
	else:
		tmp = (m + ((m * m) * (m + -2.0))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-20)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(Float64(m + Float64(Float64(m * m) * Float64(m + -2.0))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-20)
		tmp = -1.0 + (m / v);
	else
		tmp = (m + ((m * m) * (m + -2.0))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-20], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m + N[(N[(m * m), $MachinePrecision] * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.9999999999999999e-20

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. div-inv 99.9%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. *-lft-identity 99.9%

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

      5. associate-*l/ 100.0%

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

      6. *-lft-identity 100.0%

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

      7. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 100.0%

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

   if 4.9999999999999999e-20 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/r/ 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-rgt-in 54.1%

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

      4. *-un-lft-identity 54.1%

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

      5. div-inv 54.1%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in v around 0 99.9%

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

   7. Taylor expanded in m around 0 51.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot {m}^{2} + \left({m}^{3} + m\right)}}{v} \]
   8. Step-by-step derivation

      1. +-commutative 51.8%

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

      2. +-commutative 51.8%

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

      3. associate-+l+ 51.8%

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

      4. cube-mult 51.8%

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

      5. unpow2 51.8%

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

      6. distribute-rgt-out 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-20}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m + \left(m \cdot m\right) \cdot \left(m + -2\right)}{v}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. sub-neg 100.0%

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

   3. associate-/l* 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 5: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.45:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m + -2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.45) (+ -1.0 (/ m v)) (* m (* m (/ (+ m -2.0) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.45) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * (m * ((m + -2.0) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.45d0) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = m * (m * ((m + (-2.0d0)) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.45) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * (m * ((m + -2.0) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.45:
		tmp = -1.0 + (m / v)
	else:
		tmp = m * (m * ((m + -2.0) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.45)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(m * Float64(m * Float64(Float64(m + -2.0) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.45)
		tmp = -1.0 + (m / v);
	else
		tmp = m * (m * ((m + -2.0) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.45], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(N[(m + -2.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.4500000000000002

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. div-inv 99.9%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 98.1%

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

      1. sub-neg 98.1%

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

      2. *-commutative 98.1%

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

      3. distribute-rgt-in 98.1%

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

      4. *-lft-identity 98.1%

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

      5. associate-*l/ 98.2%

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

      6. *-lft-identity 98.2%

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

      7. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in v around 0 98.2%

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

   if 2.4500000000000002 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 51.1%

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

      4. *-un-lft-identity 51.1%

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

      5. div-inv 51.1%

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

      6. fma-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 99.9%

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

   7. Taylor expanded in m around inf 47.9%

      \[\leadsto \frac{\color{blue}{-2 \cdot {m}^{2} + {m}^{3}}}{v} \]
   8. Step-by-step derivation

      1. +-commutative 47.9%

         \[\leadsto \frac{\color{blue}{{m}^{3} + -2 \cdot {m}^{2}}}{v} \]

      2. cube-mult 47.9%

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

      3. unpow2 47.9%

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

      4. distribute-rgt-out 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in m around 0 19.9%

       \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   11. Step-by-step derivation

       1. +-commutative 19.9%

          \[\leadsto \color{blue}{\frac{{m}^{3}}{v} + -2 \cdot \frac{{m}^{2}}{v}} \]

       2. unpow3 19.9%

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

       3. associate-*l/ 19.9%

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

       4. associate-*r/ 19.9%

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

       5. *-commutative 19.9%

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

       6. unpow2 19.9%

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

       7. associate-*r/ 19.9%

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

       8. distribute-lft-in 99.1%

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

       9. associate-*l* 99.1%

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

       10. associate-*l/ 99.1%

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

       11. associate-*r/ 99.1%

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

   12. Simplified 99.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.45:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m + -2}{v}\right)\\ \end{array} \]

Alternative 6: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.62:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m + -2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.62) (* (- 1.0 m) (+ -1.0 (/ m v))) (* m (* m (/ (+ m -2.0) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.62) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * (m * ((m + -2.0) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.62d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (m / v))
    else
        tmp = m * (m * ((m + (-2.0d0)) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.62) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * (m * ((m + -2.0) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.62:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = m * (m * ((m + -2.0) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.62)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(m * Float64(Float64(m + -2.0) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.62)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	else
		tmp = m * (m * ((m + -2.0) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.62], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(N[(m + -2.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6200000000000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 98.3%

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

   if 1.6200000000000001 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 51.1%

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

      4. *-un-lft-identity 51.1%

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

      5. div-inv 51.1%

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

      6. fma-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 99.9%

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

   7. Taylor expanded in m around inf 47.9%

      \[\leadsto \frac{\color{blue}{-2 \cdot {m}^{2} + {m}^{3}}}{v} \]
   8. Step-by-step derivation

      1. +-commutative 47.9%

         \[\leadsto \frac{\color{blue}{{m}^{3} + -2 \cdot {m}^{2}}}{v} \]

      2. cube-mult 47.9%

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

      3. unpow2 47.9%

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

      4. distribute-rgt-out 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in m around 0 19.9%

       \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   11. Step-by-step derivation

       1. +-commutative 19.9%

          \[\leadsto \color{blue}{\frac{{m}^{3}}{v} + -2 \cdot \frac{{m}^{2}}{v}} \]

       2. unpow3 19.9%

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

       3. associate-*l/ 19.9%

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

       4. associate-*r/ 19.9%

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

       5. *-commutative 19.9%

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

       6. unpow2 19.9%

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

       7. associate-*r/ 19.9%

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

       8. distribute-lft-in 99.1%

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

       9. associate-*l* 99.1%

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

       10. associate-*l/ 99.1%

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

       11. associate-*r/ 99.1%

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

   12. Simplified 99.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.62:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m + -2}{v}\right)\\ \end{array} \]

Alternative 7: 72.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-114}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.6e-114) -1.0 (if (<= m 0.28) (/ m v) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.6e-114) {
		tmp = -1.0;
	} else if (m <= 0.28) {
		tmp = m / v;
	} else {
		tmp = m * (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 (m <= 3.6d-114) then
        tmp = -1.0d0
    else if (m <= 0.28d0) then
        tmp = m / v
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.6e-114) {
		tmp = -1.0;
	} else if (m <= 0.28) {
		tmp = m / v;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.6e-114:
		tmp = -1.0
	elif m <= 0.28:
		tmp = m / v
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.6e-114)
		tmp = -1.0;
	elseif (m <= 0.28)
		tmp = Float64(m / v);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.6e-114)
		tmp = -1.0;
	elseif (m <= 0.28)
		tmp = m / v;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.6e-114], -1.0, If[LessEqual[m, 0.28], N[(m / v), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 3.60000000000000018e-114

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 76.7%

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

   if 3.60000000000000018e-114 < m < 0.28000000000000003

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 94.5%

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

      1. distribute-lft-in 94.5%

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

      2. sub-neg 94.5%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 94.2%

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

      5. sqr-neg 94.2%

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

      6. sqrt-unprod 94.2%

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

      7. add-sqr-sqrt 94.2%

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

      8. sub-neg 94.2%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 94.2%

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

      11. sqr-neg 94.2%

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

      12. sqrt-unprod 94.2%

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

      13. add-sqr-sqrt 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. distribute-lft-in 94.2%

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

      2. +-commutative 94.2%

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

      3. +-commutative 94.2%

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

   8. Simplified 94.2%

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

   9. Taylor expanded in v around 0 83.0%

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

   10. Taylor expanded in m around 0 83.1%

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

   if 0.28000000000000003 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 0.1%

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

      1. distribute-lft-in 0.1%

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

      2. sub-neg 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 81.0%

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

      5. sqr-neg 81.0%

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

      6. sqrt-unprod 81.0%

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

      7. add-sqr-sqrt 81.0%

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

      8. sub-neg 81.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 29.8%

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

      11. sqr-neg 29.8%

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

      12. sqrt-unprod 81.0%

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

      13. add-sqr-sqrt 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. distribute-lft-in 81.0%

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

      2. +-commutative 81.0%

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

      3. +-commutative 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in m around inf 81.0%

      \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} \]
   10. Step-by-step derivation

       1. unpow2 81.0%

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

       2. associate-*r/ 81.0%

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

   11. Simplified 81.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-114}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 8: 87.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (+ -1.0 (/ m v)) (* m (/ (+ m 1.0) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.4d0) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = m * ((m + 1.0d0) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = -1.0 + (m / v)
	else:
		tmp = m * ((m + 1.0) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(m * Float64(Float64(m + 1.0) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = -1.0 + (m / v);
	else
		tmp = m * ((m + 1.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m + 1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. div-inv 99.9%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 98.1%

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

      1. sub-neg 98.1%

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

      2. *-commutative 98.1%

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

      3. distribute-rgt-in 98.1%

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

      4. *-lft-identity 98.1%

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

      5. associate-*l/ 98.2%

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

      6. *-lft-identity 98.2%

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

      7. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in v around 0 98.2%

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

   if 2.39999999999999991 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 0.1%

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

      1. distribute-lft-in 0.1%

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

      2. sub-neg 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 81.0%

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

      5. sqr-neg 81.0%

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

      6. sqrt-unprod 81.0%

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

      7. add-sqr-sqrt 81.0%

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

      8. sub-neg 81.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 29.8%

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

      11. sqr-neg 29.8%

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

      12. sqrt-unprod 81.0%

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

      13. add-sqr-sqrt 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. distribute-lft-in 81.0%

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

      2. +-commutative 81.0%

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

      3. +-commutative 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in v around 0 81.0%

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

       1. div-inv 81.0%

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

       2. associate-*l* 81.0%

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

       3. add-sqr-sqrt 81.0%

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

       4. sqrt-prod 81.0%

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

       5. sqr-neg 81.0%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 0.1%

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

       8. sub-neg 0.1%

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

       9. div-inv 0.1%

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

       10. *-un-lft-identity 0.1%

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

       11. *-un-lft-identity 0.1%

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

       12. sub-neg 0.1%

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

       13. add-sqr-sqrt 0.0%

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

       14. sqrt-unprod 81.0%

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

       15. sqr-neg 81.0%

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

       16. sqrt-prod 81.0%

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

       17. add-sqr-sqrt 81.0%

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

       18. +-commutative 81.0%

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

   11. Applied egg-rr 81.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \]

Alternative 9: 87.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.28:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.28) (+ -1.0 (/ m v)) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.28) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * (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 (m <= 0.28d0) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.28) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.28:
		tmp = -1.0 + (m / v)
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.28)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.28)
		tmp = -1.0 + (m / v);
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.28], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.28000000000000003

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. div-inv 99.9%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 98.1%

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

      1. sub-neg 98.1%

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

      2. *-commutative 98.1%

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

      3. distribute-rgt-in 98.1%

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

      4. *-lft-identity 98.1%

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

      5. associate-*l/ 98.2%

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

      6. *-lft-identity 98.2%

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

      7. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in v around 0 98.2%

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

   if 0.28000000000000003 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 0.1%

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

      1. distribute-lft-in 0.1%

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

      2. sub-neg 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 81.0%

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

      5. sqr-neg 81.0%

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

      6. sqrt-unprod 81.0%

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

      7. add-sqr-sqrt 81.0%

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

      8. sub-neg 81.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 29.8%

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

      11. sqr-neg 29.8%

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

      12. sqrt-unprod 81.0%

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

      13. add-sqr-sqrt 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. distribute-lft-in 81.0%

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

      2. +-commutative 81.0%

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

      3. +-commutative 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in m around inf 81.0%

      \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} \]
   10. Step-by-step derivation

       1. unpow2 81.0%

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

       2. associate-*r/ 81.0%

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

   11. Simplified 81.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.28:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 10: 61.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-115}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 6.5e-115) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.5e-115) {
		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 (m <= 6.5d-115) 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 (m <= 6.5e-115) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.5e-115:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.5e-115)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.5e-115)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.5e-115], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.50000000000000033e-115

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 76.7%

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

   if 6.50000000000000033e-115 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 23.4%

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

      1. distribute-lft-in 23.4%

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

      2. sub-neg 23.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 84.3%

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

      5. sqr-neg 84.3%

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

      6. sqrt-unprod 84.3%

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

      7. add-sqr-sqrt 84.3%

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

      8. sub-neg 84.3%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 45.7%

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

      11. sqr-neg 45.7%

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

      12. sqrt-unprod 84.3%

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

      13. add-sqr-sqrt 84.3%

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

   6. Applied egg-rr 84.3%

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

      1. distribute-lft-in 84.3%

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

      2. +-commutative 84.3%

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

      3. +-commutative 84.3%

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

   8. Simplified 84.3%

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

   9. Taylor expanded in v around 0 81.5%

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

   10. Taylor expanded in m around 0 60.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-115}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 11: 26.8% accurate, 4.3× 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 100.0%

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

   1. *-commutative 100.0%

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

   2. sub-neg 100.0%

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

   3. associate-/l* 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around inf 30.9%

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

   1. neg-mul-1 30.9%

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

   2. neg-sub0 30.9%

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

   3. associate--r- 30.9%

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

   4. metadata-eval 30.9%

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

6. Simplified 30.9%

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

7. Final simplification 30.9%

   \[\leadsto m + -1 \]

Alternative 12: 24.4% accurate, 13.0× 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 100.0%

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

   1. *-commutative 100.0%

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

   2. sub-neg 100.0%

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

   3. associate-/l* 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in m around 0 28.4%

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

5. Final simplification 28.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023274 
(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)))