b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.1s

Alternatives: 12

Speedup: 1.0×

• 40.8% 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

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

FPCore

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

C

double code(double m, double v) {
	return (1.0 - m) * (((1.0 - m) * (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) * (((1.0d0 - m) * (m / v)) + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\frac{m}{\frac{v}{1 - m}} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6)
   (+ (/ m (/ v (- 1.0 m))) (+ m -1.0))
   (* (* m (/ m v)) (+ m -2.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (m / (v / (1.0 - m))) + (m + -1.0);
	} else {
		tmp = (m * (m / v)) * (m + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (m / (v / (1.0d0 - m))) + (m + (-1.0d0))
    else
        tmp = (m * (m / v)) * (m + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (m / (v / (1.0 - m))) + (m + -1.0);
	} else {
		tmp = (m * (m / v)) * (m + -2.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (m / (v / (1.0 - m))) + (m + -1.0)
	else:
		tmp = (m * (m / v)) * (m + -2.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(m / Float64(v / Float64(1.0 - m))) + Float64(m + -1.0));
	else
		tmp = Float64(Float64(m * Float64(m / v)) * Float64(m + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (m / (v / (1.0 - m))) + (m + -1.0);
	else
		tmp = (m * (m / v)) * (m + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(m + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 98.3%

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

   3. Taylor expanded in v around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. mul-1-neg 98.3%

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

      3. unsub-neg 98.3%

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

      4. associate-/l* 98.3%

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

   5. Simplified 98.3%

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

   if 1.6000000000000001 < 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-lft-in 49.5%

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

      4. *-commutative 49.5%

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

      5. *-un-lft-identity 49.5%

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

      6. frac-2neg 49.5%

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

      7. div-inv 49.5%

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

      8. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-udef 49.5%

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

      2. *-commutative 49.5%

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

      3. distribute-lft-out 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-*r/ 99.9%

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

      6. neg-mul-1 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. metadata-eval 99.9%

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

      9. sub-neg 99.9%

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

      10. associate-/r* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf 23.5%

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

       1. *-commutative 23.5%

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

       2. unpow2 23.5%

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

       3. unpow3 23.5%

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

       4. associate-*l/ 23.5%

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

       5. distribute-lft-out 99.1%

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

       6. associate-*r/ 99.1%

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

   13. Simplified 99.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\frac{m}{\frac{v}{1 - m}} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.2 \cdot 10^{-18}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{\frac{v}{m}}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.2e-18)
   (+ -1.0 (+ m (/ m v)))
   (/ (- 1.0 m) (/ (/ v m) (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.2e-18) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (1.0 - m) / ((v / m) / (1.0 - m));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 4.2d-18) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (1.0d0 - m) / ((v / m) / (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.2e-18) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (1.0 - m) / ((v / m) / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.2e-18:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (1.0 - m) / ((v / m) / (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.2e-18)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) / Float64(Float64(v / m) / Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.2e-18)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (1.0 - m) / ((v / m) / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.2e-18], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] / N[(N[(v / m), $MachinePrecision] / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.19999999999999999e-18

   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 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

   6. Simplified 100.0%

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

   if 4.19999999999999999e-18 < 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-lft-in 52.3%

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

      4. *-commutative 52.3%

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

      5. *-un-lft-identity 52.3%

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

      6. frac-2neg 52.3%

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

      7. div-inv 52.3%

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

      8. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-udef 52.3%

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

      2. *-commutative 52.3%

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

      3. distribute-lft-out 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in v around 0 99.8%

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

      1. associate-/l* 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-*r/ 99.7%

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

      6. neg-mul-1 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. associate-/r* 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.2 \cdot 10^{-18}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{\frac{v}{m}}{1 - m}}\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-31}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.9e-31)
   (+ -1.0 (+ m (/ m v)))
   (/ (* (- 1.0 m) (* m (- 1.0 m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-31) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = ((1.0 - m) * (m * (1.0 - 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 <= 2.9d-31) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = ((1.0d0 - m) * (m * (1.0d0 - m))) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-31) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = ((1.0 - m) * (m * (1.0 - m))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.9e-31:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = ((1.0 - m) * (m * (1.0 - m))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.9e-31)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(m * Float64(1.0 - m))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.9e-31)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = ((1.0 - m) * (m * (1.0 - m))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.9e-31], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.9000000000000001e-31

   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 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

   6. Simplified 100.0%

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

   if 2.9000000000000001e-31 < 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-lft-in 54.8%

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

      4. *-commutative 54.8%

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

      5. *-un-lft-identity 54.8%

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

      6. frac-2neg 54.8%

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

      7. div-inv 54.7%

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

      8. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-udef 54.7%

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

      2. *-commutative 54.7%

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

      3. distribute-lft-out 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in v around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. associate-*r* 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

      5. +-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. +-commutative 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-31}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \]

Alternative 5: 84.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.05 \cdot 10^{-165}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.05e-165) -1.0 (if (<= m 0.38) (/ m v) (* (/ m v) (* m m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.05e-165) {
		tmp = -1.0;
	} else if (m <= 0.38) {
		tmp = m / v;
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 4.05d-165) then
        tmp = -1.0d0
    else if (m <= 0.38d0) then
        tmp = m / v
    else
        tmp = (m / v) * (m * m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.05e-165) {
		tmp = -1.0;
	} else if (m <= 0.38) {
		tmp = m / v;
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.05e-165:
		tmp = -1.0
	elif m <= 0.38:
		tmp = m / v
	else:
		tmp = (m / v) * (m * m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.05e-165)
		tmp = -1.0;
	elseif (m <= 0.38)
		tmp = Float64(m / v);
	else
		tmp = Float64(Float64(m / v) * Float64(m * m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.05e-165)
		tmp = -1.0;
	elseif (m <= 0.38)
		tmp = m / v;
	else
		tmp = (m / v) * (m * m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.05e-165], -1.0, If[LessEqual[m, 0.38], N[(m / v), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 4.05000000000000012e-165

   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 75.6%

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

   if 4.05000000000000012e-165 < m < 0.38

   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-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. *-un-lft-identity 99.9%

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

      6. frac-2neg 99.9%

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

      7. div-inv 99.5%

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

      8. fma-def 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. fma-udef 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-lft-out 99.6%

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

      4. neg-mul-1 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in v around 0 80.7%

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

      1. associate-/l* 80.4%

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

      2. sub-neg 80.4%

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

      3. metadata-eval 80.4%

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

      4. +-commutative 80.4%

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

      5. associate-*r/ 80.4%

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

      6. neg-mul-1 80.4%

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

      7. distribute-neg-in 80.4%

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

      8. metadata-eval 80.4%

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

      9. sub-neg 80.4%

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

      10. associate-/r* 80.4%

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

   10. Simplified 80.4%

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

   11. Taylor expanded in m around 0 77.5%

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

   if 0.38 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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 inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unpow2 98.7%

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

      3. associate-*l/ 98.7%

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

      4. distribute-rgt-neg-out 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in v around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unpow2 98.7%

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

      3. associate-/l* 98.7%

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

      4. distribute-neg-frac 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in m around inf 98.7%

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

       1. associate-*r/ 98.7%

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

       2. neg-mul-1 98.7%

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

   12. Simplified 98.7%

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

       1. div-inv 98.6%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 0.1%

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

       4. sqr-neg 0.1%

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

       5. sqrt-unprod 0.1%

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

       6. add-sqr-sqrt 0.1%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 96.2%

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

       9. sqr-neg 96.2%

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

       10. sqrt-unprod 98.6%

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

       11. add-sqr-sqrt 98.6%

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

       12. clear-num 98.7%

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

   14. Applied egg-rr 98.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.05 \cdot 10^{-165}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]

Alternative 6: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.5:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.5) (+ -1.0 (+ m (/ m v))) (* (* m (/ m v)) (+ m -2.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.5) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (m / v)) * (m + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.5d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m * (m / v)) * (m + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.5) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (m / v)) * (m + -2.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.5:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m * (m / v)) * (m + -2.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.5)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(m / v)) * Float64(m + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.5)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m * (m / v)) * (m + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.5], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5

   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 98.0%

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

      1. sub-neg 98.0%

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

      2. metadata-eval 98.0%

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

      3. +-commutative 98.0%

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

      4. *-commutative 98.0%

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

      5. distribute-lft-in 98.0%

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

      6. *-rgt-identity 98.0%

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

      7. associate-*r/ 98.2%

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

      8. *-rgt-identity 98.2%

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

   6. Simplified 98.2%

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

   if 2.5 < 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-lft-in 49.5%

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

      4. *-commutative 49.5%

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

      5. *-un-lft-identity 49.5%

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

      6. frac-2neg 49.5%

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

      7. div-inv 49.5%

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

      8. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-udef 49.5%

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

      2. *-commutative 49.5%

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

      3. distribute-lft-out 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-*r/ 99.9%

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

      6. neg-mul-1 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. metadata-eval 99.9%

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

      9. sub-neg 99.9%

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

      10. associate-/r* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf 23.5%

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

       1. *-commutative 23.5%

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

       2. unpow2 23.5%

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

       3. unpow3 23.5%

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

       4. associate-*l/ 23.5%

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

       5. distribute-lft-out 99.1%

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

       6. associate-*r/ 99.1%

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

   13. Simplified 99.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.5:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- 1.0 m) (+ (/ m v) -1.0)) (* (* m (/ m v)) (+ m -2.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m * (m / v)) * (m + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = (m * (m / v)) * (m + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m * (m / v)) * (m + -2.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = (m * (m / v)) * (m + -2.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m * Float64(m / v)) * Float64(m + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = (m * (m / v)) * (m + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 98.3%

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

   if 1.6000000000000001 < 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-lft-in 49.5%

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

      4. *-commutative 49.5%

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

      5. *-un-lft-identity 49.5%

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

      6. frac-2neg 49.5%

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

      7. div-inv 49.5%

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

      8. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-udef 49.5%

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

      2. *-commutative 49.5%

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

      3. distribute-lft-out 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-*r/ 99.9%

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

      6. neg-mul-1 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. metadata-eval 99.9%

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

      9. sub-neg 99.9%

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

      10. associate-/r* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf 23.5%

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

       1. *-commutative 23.5%

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

       2. unpow2 23.5%

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

       3. unpow3 23.5%

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

       4. associate-*l/ 23.5%

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

       5. distribute-lft-out 99.1%

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

       6. associate-*r/ 99.1%

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

   13. Simplified 99.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \]

Alternative 8: 73.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.95 \cdot 10^{-164}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.95e-164) -1.0 (if (<= m 0.28) (/ m v) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.95e-164) {
		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 <= 1.95d-164) 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 <= 1.95e-164) {
		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 <= 1.95e-164:
		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 <= 1.95e-164)
		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 <= 1.95e-164)
		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, 1.95e-164], -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 < 1.9499999999999999e-164

   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 75.6%

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

   if 1.9499999999999999e-164 < 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. 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-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. *-un-lft-identity 99.9%

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

      6. frac-2neg 99.9%

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

      7. div-inv 99.5%

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

      8. fma-def 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. fma-udef 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-lft-out 99.6%

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

      4. neg-mul-1 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in v around 0 80.7%

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

      1. associate-/l* 80.4%

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

      2. sub-neg 80.4%

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

      3. metadata-eval 80.4%

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

      4. +-commutative 80.4%

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

      5. associate-*r/ 80.4%

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

      6. neg-mul-1 80.4%

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

      7. distribute-neg-in 80.4%

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

      8. metadata-eval 80.4%

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

      9. sub-neg 80.4%

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

      10. associate-/r* 80.4%

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

   10. Simplified 80.4%

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

   11. Taylor expanded in m around 0 77.5%

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

   if 0.28000000000000003 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 0.1%

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

      1. sub-neg 0.1%

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

      2. distribute-lft-in 0.1%

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

      3. *-commutative 0.1%

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

      4. *-un-lft-identity 0.1%

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

      5. sub-neg 0.1%

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

      6. metadata-eval 0.1%

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

      7. sub-neg 0.1%

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

      8. metadata-eval 0.1%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 77.9%

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

      11. sqr-neg 77.9%

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

      12. sqrt-unprod 77.9%

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

      13. add-sqr-sqrt 77.9%

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

   4. Applied egg-rr 77.9%

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

      1. *-commutative 77.9%

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

      2. distribute-rgt1-in 77.9%

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

   6. Simplified 77.9%

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

   7. Taylor expanded in m around inf 77.9%

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

      1. unpow2 77.9%

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

      2. associate-*r/ 77.9%

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

   9. Simplified 77.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.95 \cdot 10^{-164}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 9: 97.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 0.38d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m / v) * (m * m)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.38

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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 98.0%

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

      1. sub-neg 98.0%

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

      2. metadata-eval 98.0%

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

      3. +-commutative 98.0%

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

      4. *-commutative 98.0%

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

      5. distribute-lft-in 98.0%

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

      6. *-rgt-identity 98.0%

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

      7. associate-*r/ 98.2%

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

      8. *-rgt-identity 98.2%

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

   6. Simplified 98.2%

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

   if 0.38 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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 inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unpow2 98.7%

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

      3. associate-*l/ 98.7%

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

      4. distribute-rgt-neg-out 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in v around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unpow2 98.7%

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

      3. associate-/l* 98.7%

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

      4. distribute-neg-frac 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in m around inf 98.7%

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

       1. associate-*r/ 98.7%

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

       2. neg-mul-1 98.7%

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

   12. Simplified 98.7%

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

       1. div-inv 98.6%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 0.1%

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

       4. sqr-neg 0.1%

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

       5. sqrt-unprod 0.1%

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

       6. add-sqr-sqrt 0.1%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 96.2%

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

       9. sqr-neg 96.2%

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

       10. sqrt-unprod 98.6%

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

       11. add-sqr-sqrt 98.6%

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

       12. clear-num 98.7%

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

   14. Applied egg-rr 98.7%

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

4. Final simplification 98.4%

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

Alternative 10: 62.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9 \cdot 10^{-164}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 9e-164) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 9e-164) {
		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 <= 9d-164) 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 <= 9e-164) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 9e-164:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 9e-164)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 9e-164)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 9e-164], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.9999999999999995e-164

   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 75.6%

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

   if 8.9999999999999995e-164 < 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-lft-in 68.0%

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

      4. *-commutative 68.0%

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

      5. *-un-lft-identity 68.0%

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

      6. frac-2neg 68.0%

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

      7. div-inv 67.9%

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

      8. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-udef 67.9%

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

      2. *-commutative 67.9%

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

      3. distribute-lft-out 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in v around 0 92.9%

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

      1. associate-/l* 92.8%

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

      2. sub-neg 92.8%

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

      3. metadata-eval 92.8%

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

      4. +-commutative 92.8%

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

      5. associate-*r/ 92.8%

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

      6. neg-mul-1 92.8%

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

      7. distribute-neg-in 92.8%

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

      8. metadata-eval 92.8%

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

      9. sub-neg 92.8%

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

      10. associate-/r* 92.8%

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

   10. Simplified 92.8%

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

   11. Taylor expanded in m around 0 62.6%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9 \cdot 10^{-164}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 11: 27.5% 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 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 v around inf 26.8%

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

   1. neg-mul-1 26.8%

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

   2. neg-sub0 26.8%

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

   3. associate--r- 26.8%

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

   4. metadata-eval 26.8%

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

6. Simplified 26.8%

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

7. Final simplification 26.8%

   \[\leadsto m + -1 \]

Alternative 12: 25.1% 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 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 24.6%

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

5. Final simplification 24.6%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023268 
(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)))