b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.6s

Alternatives: 13

Speedup: N/A×

• 41.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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* 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 98.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 97.0%

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

      1. distribute-rgt-in 97.0%

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

      2. *-un-lft-identity 97.0%

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

      3. metadata-eval 97.0%

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

      4. associate-/r/ 97.0%

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

      5. div-inv 97.0%

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

      6. associate-/l/ 97.0%

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

      7. associate-*l/ 97.0%

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

      8. *-un-lft-identity 97.0%

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

      9. mul-1-neg 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in m around 0 97.0%

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

      1. distribute-rgt1-in 97.0%

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

      2. +-commutative 97.0%

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

      3. *-commutative 97.0%

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

      4. +-commutative 97.0%

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

   9. Simplified 97.0%

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

   10. Taylor expanded in v around 0 97.0%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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 97.4%

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

      1. mul-1-neg 97.4%

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

      2. distribute-neg-frac 97.4%

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

   6. Simplified 97.4%

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

4. Final simplification 97.2%

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

Alternative 3: 98.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 97.0%

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

      1. distribute-rgt-in 97.0%

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

      2. *-un-lft-identity 97.0%

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

      3. metadata-eval 97.0%

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

      4. associate-/r/ 97.0%

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

      5. div-inv 97.0%

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

      6. associate-/l/ 97.0%

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

      7. associate-*l/ 97.0%

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

      8. *-un-lft-identity 97.0%

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

      9. mul-1-neg 97.0%

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

   6. Applied egg-rr 97.0%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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 97.4%

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

      1. mul-1-neg 97.4%

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

      2. distribute-neg-frac 97.4%

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

   6. Simplified 97.4%

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

4. Final simplification 97.2%

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

Alternative 4: 87.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 97.0%

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

      1. distribute-rgt-in 97.0%

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

      2. *-un-lft-identity 97.0%

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

      3. metadata-eval 97.0%

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

      4. associate-/r/ 97.0%

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

      5. div-inv 97.0%

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

      6. associate-/l/ 97.0%

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

      7. associate-*l/ 97.0%

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

      8. *-un-lft-identity 97.0%

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

      9. mul-1-neg 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in m around 0 97.0%

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

      1. distribute-rgt1-in 97.0%

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

      2. +-commutative 97.0%

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

      3. *-commutative 97.0%

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

      4. +-commutative 97.0%

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

   9. Simplified 97.0%

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

   10. Taylor expanded in v around 0 97.0%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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. div-inv 99.9%

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

      2. *-commutative 99.9%

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

      3. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-/r/ 0.1%

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

      3. distribute-rgt-in 0.1%

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

      4. neg-mul-1 0.1%

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

      5. neg-sub0 0.1%

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

      6. sub-neg 0.1%

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

      7. add-sqr-sqrt 0.1%

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

      8. sqrt-unprod 0.1%

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

      9. sqr-neg 0.1%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 0.1%

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

      12. associate-+l+ 0.1%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 0.1%

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

      15. sqr-neg 0.1%

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

      16. sqrt-unprod 0.1%

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

      17. add-sqr-sqrt 0.1%

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

      18. neg-mul-1 0.1%

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

      19. distribute-rgt-in 0.1%

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

   8. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{0 + \left(m + -1\right) \cdot \left(-1 + \frac{m}{v}\right)} \]
   9. Step-by-step derivation

      1. +-lft-identity 83.3%

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

      2. +-commutative 83.3%

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

   10. Simplified 83.3%

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

4. Final simplification 90.3%

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

Alternative 5: 87.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 97.0%

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

      1. distribute-rgt-in 97.0%

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

      2. *-un-lft-identity 97.0%

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

      3. metadata-eval 97.0%

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

      4. associate-/r/ 97.0%

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

      5. div-inv 97.0%

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

      6. associate-/l/ 97.0%

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

      7. associate-*l/ 97.0%

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

      8. *-un-lft-identity 97.0%

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

      9. mul-1-neg 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in m around 0 97.0%

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

      1. distribute-rgt1-in 97.0%

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

      2. +-commutative 97.0%

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

      3. *-commutative 97.0%

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

      4. +-commutative 97.0%

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

   9. Simplified 97.0%

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

   10. Taylor expanded in v around 0 97.0%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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. div-inv 99.9%

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

      2. *-commutative 99.9%

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

      3. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. associate-/r/ 0.1%

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

      2. distribute-rgt-in 0.1%

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

      3. neg-mul-1 0.1%

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

      4. add-sqr-sqrt 0.1%

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

      5. sqrt-unprod 0.1%

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

      6. sqr-neg 0.1%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.1%

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

      9. expm1-log1p-u 0.0%

         \[\leadsto \frac{1}{\frac{v}{m}} \cdot \left(1 - m\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]

      10. expm1-udef 0.0%

          \[\leadsto \frac{1}{\frac{v}{m}} \cdot \left(1 - m\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \]

      11. associate-+r- 0.0%

          \[\leadsto \color{blue}{\left(\frac{1}{\frac{v}{m}} \cdot \left(1 - m\right) + e^{\mathsf{log1p}\left(1 - m\right)}\right) - 1} \]

   8. Applied egg-rr 83.3%

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

4. Final simplification 90.3%

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

Alternative 6: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 97.0%

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

      1. distribute-rgt-in 97.0%

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

      2. *-un-lft-identity 97.0%

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

      3. metadata-eval 97.0%

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

      4. associate-/r/ 97.0%

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

      5. div-inv 97.0%

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

      6. associate-/l/ 97.0%

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

      7. associate-*l/ 97.0%

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

      8. *-un-lft-identity 97.0%

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

      9. mul-1-neg 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in m around 0 97.0%

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

      1. distribute-rgt1-in 97.0%

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

      2. +-commutative 97.0%

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

      3. *-commutative 97.0%

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

      4. +-commutative 97.0%

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

   9. Simplified 97.0%

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

   10. Taylor expanded in v around 0 97.0%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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 97.4%

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

      1. mul-1-neg 97.4%

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

      2. distribute-neg-frac 97.4%

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

   6. Simplified 97.4%

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

      1. frac-2neg 97.4%

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

      2. associate-/r/ 97.3%

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

      3. remove-double-neg 97.3%

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

   8. Applied egg-rr 97.3%

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

4. Final simplification 97.2%

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

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. 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/ 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 8: 87.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 96.6%

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

      1. distribute-rgt-in 96.6%

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

      2. *-un-lft-identity 96.6%

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

      3. +-commutative 96.6%

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

      4. *-commutative 96.6%

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

      5. div-inv 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. add-sqr-sqrt 96.9%

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

      2. sqrt-unprod 96.9%

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

      3. sqr-neg 96.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 96.9%

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

      6. sub-neg 96.9%

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

   8. Applied egg-rr 96.9%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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. div-inv 99.9%

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

      2. *-commutative 99.9%

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

      3. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-/r/ 0.1%

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

      3. distribute-rgt-in 0.1%

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

      4. neg-mul-1 0.1%

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

      5. neg-sub0 0.1%

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

      6. sub-neg 0.1%

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

      7. add-sqr-sqrt 0.1%

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

      8. sqrt-unprod 0.1%

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

      9. sqr-neg 0.1%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 0.1%

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

      12. associate-+l+ 0.1%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 0.1%

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

      15. sqr-neg 0.1%

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

      16. sqrt-unprod 0.1%

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

      17. add-sqr-sqrt 0.1%

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

      18. neg-mul-1 0.1%

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

      19. distribute-rgt-in 0.1%

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

   8. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{0 + \left(m + -1\right) \cdot \left(-1 + \frac{m}{v}\right)} \]
   9. Step-by-step derivation

      1. +-lft-identity 83.3%

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

      2. +-commutative 83.3%

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

   10. Simplified 83.3%

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

4. Final simplification 90.3%

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

Alternative 9: 87.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{m}{v}\\ \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\left(m + -1\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ m v))))
   (if (<= m 1.0) (* (- 1.0 m) t_0) (* (+ m -1.0) t_0))))

C

double code(double m, double v) {
	double t_0 = -1.0 + (m / v);
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * t_0;
	} else {
		tmp = (m + -1.0) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + (m / v)
    if (m <= 1.0d0) then
        tmp = (1.0d0 - m) * t_0
    else
        tmp = (m + (-1.0d0)) * t_0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = -1.0 + (m / v);
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * t_0;
	} else {
		tmp = (m + -1.0) * t_0;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = -1.0 + (m / v)
	tmp = 0
	if m <= 1.0:
		tmp = (1.0 - m) * t_0
	else:
		tmp = (m + -1.0) * t_0
	return tmp

Julia

function code(m, v)
	t_0 = Float64(-1.0 + Float64(m / v))
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(1.0 - m) * t_0);
	else
		tmp = Float64(Float64(m + -1.0) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = -1.0 + (m / v);
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (1.0 - m) * t_0;
	else
		tmp = (m + -1.0) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(m + -1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   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/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 97.0%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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. div-inv 99.9%

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

      2. *-commutative 99.9%

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

      3. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-/r/ 0.1%

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

      3. distribute-rgt-in 0.1%

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

      4. neg-mul-1 0.1%

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

      5. neg-sub0 0.1%

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

      6. sub-neg 0.1%

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

      7. add-sqr-sqrt 0.1%

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

      8. sqrt-unprod 0.1%

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

      9. sqr-neg 0.1%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 0.1%

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

      12. associate-+l+ 0.1%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 0.1%

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

      15. sqr-neg 0.1%

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

      16. sqrt-unprod 0.1%

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

      17. add-sqr-sqrt 0.1%

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

      18. neg-mul-1 0.1%

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

      19. distribute-rgt-in 0.1%

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

   8. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{0 + \left(m + -1\right) \cdot \left(-1 + \frac{m}{v}\right)} \]
   9. Step-by-step derivation

      1. +-lft-identity 83.3%

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

      2. +-commutative 83.3%

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

   10. Simplified 83.3%

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

4. Final simplification 90.3%

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

Alternative 10: 75.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. 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/ 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in m around 0 80.0%

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

   1. distribute-rgt-in 80.0%

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

   2. *-un-lft-identity 80.0%

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

   3. +-commutative 80.0%

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

   4. *-commutative 80.0%

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

   5. div-inv 80.1%

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

6. Applied egg-rr 80.1%

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

7. Final simplification 80.1%

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

Alternative 11: 26.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.8 \cdot 10^{-66}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 4.8e-66) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.8e-66) {
		tmp = -1.0;
	} else {
		tmp = 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 <= 4.8d-66) then
        tmp = -1.0d0
    else
        tmp = m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.8e-66) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.8e-66:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.8e-66)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.8e-66)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.8e-66], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 4.80000000000000052e-66

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 60.8%

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

   if 4.80000000000000052e-66 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 19.6%

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

      1. distribute-rgt-in 19.6%

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

      2. *-un-lft-identity 19.6%

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

      3. metadata-eval 19.6%

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

      4. associate-/r/ 19.6%

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

      5. div-inv 19.6%

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

      6. associate-/l/ 19.6%

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

      7. associate-*l/ 19.6%

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

      8. *-un-lft-identity 19.6%

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

      9. mul-1-neg 19.6%

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

   6. Applied egg-rr 19.6%

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

   7. Taylor expanded in m around 0 25.3%

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

      1. distribute-rgt1-in 25.3%

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

      2. +-commutative 25.3%

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

      3. *-commutative 25.3%

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

      4. +-commutative 25.3%

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

   9. Simplified 25.3%

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

   10. Taylor expanded in m around inf 5.7%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.8 \cdot 10^{-66}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]

Alternative 12: 27.2% 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/ 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around inf 26.2%

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

   1. sub-neg 26.2%

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

   2. distribute-lft-in 26.2%

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

   3. metadata-eval 26.2%

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

   4. mul-1-neg 26.2%

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

   5. remove-double-neg 26.2%

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

6. Simplified 26.2%

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

7. Final simplification 26.2%

   \[\leadsto m + -1 \]

Alternative 13: 24.7% 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/ 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in m around 0 23.6%

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

5. Final simplification 23.6%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023301 
(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)))