b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

Alternatives: 14

Speedup: 1.0×

• 41.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. *-commutative 99.9%

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

   2. flip-- 99.9%

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

   3. associate-*r/ 97.5%

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

   4. fma-def 97.5%

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

   5. metadata-eval 97.5%

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

   6. +-commutative 97.5%

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

5. Applied egg-rr 97.5%

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

   1. associate-/l* 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double t_0 = v / (1.0 - m);
	double tmp;
	if (m <= 1.0) {
		tmp = (m / t_0) + (m + -1.0);
	} else {
		tmp = (m * -m) / 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 = v / (1.0d0 - m)
    if (m <= 1.0d0) then
        tmp = (m / t_0) + (m + (-1.0d0))
    else
        tmp = (m * -m) / t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 1.0], N[(N[(m / t$95$0), $MachinePrecision] + N[(m + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m * (-m)), $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. Taylor expanded in m around 0 98.1%

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

   3. Taylor expanded in v around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. mul-1-neg 98.1%

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

      3. unsub-neg 98.1%

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

      4. associate-/l* 98.1%

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

   5. Simplified 98.1%

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

   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/ 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around inf 98.4%

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

      1. unpow2 98.4%

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

      2. associate-*l/ 98.3%

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

      3. neg-mul-1 98.3%

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

      4. distribute-rgt-neg-out 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in v around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unpow2 98.4%

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

      3. associate-/l* 98.4%

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

      4. distribute-neg-frac 98.4%

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

      5. distribute-rgt-neg-in 98.4%

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

   9. Simplified 98.4%

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

4. Final simplification 98.3%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $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/ 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. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / 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. Final simplification 99.9%

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

Alternative 5: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m * Float64(-m)) / Float64(v / Float64(1.0 - 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);
	else
		tmp = (m * -m) / (v / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m * (-m)), $MachinePrecision] / N[(v / N[(1.0 - m), $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. Taylor expanded in m around 0 98.1%

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

   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/ 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around inf 98.4%

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

      1. unpow2 98.4%

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

      2. associate-*l/ 98.3%

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

      3. neg-mul-1 98.3%

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

      4. distribute-rgt-neg-out 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in v around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unpow2 98.4%

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

      3. associate-/l* 98.4%

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

      4. distribute-neg-frac 98.4%

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

      5. distribute-rgt-neg-in 98.4%

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

   9. Simplified 98.4%

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

4. Final simplification 98.3%

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

Alternative 6: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = 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 = (m / v) + -1.0;
	else
		tmp = m * ((m / v) * (m + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

      1. *-commutative 100.0%

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

      2. flip-- 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 97.8%

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

      1. sub-neg 97.8%

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

      2. *-commutative 97.8%

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

      3. distribute-rgt-in 97.8%

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

      4. *-lft-identity 97.8%

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

      5. associate-*l/ 98.0%

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

      6. *-lft-identity 98.0%

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

      7. metadata-eval 98.0%

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

   10. Simplified 98.0%

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

   11. Taylor expanded in v around 0 98.0%

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

   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/ 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around inf 98.4%

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

      1. unpow2 98.4%

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

      2. associate-*l/ 98.3%

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

      3. neg-mul-1 98.3%

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

      4. distribute-rgt-neg-out 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in v around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unpow2 98.4%

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

      3. associate-/l* 98.4%

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

      4. distribute-neg-frac 98.4%

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

      5. distribute-rgt-neg-in 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in m around 0 23.0%

       \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   11. Step-by-step derivation

       1. +-commutative 23.0%

          \[\leadsto \color{blue}{\frac{{m}^{3}}{v} + -1 \cdot \frac{{m}^{2}}{v}} \]

       2. unpow3 23.0%

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

       3. associate-*r/ 23.0%

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

       4. associate-*r* 23.0%

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

       5. unpow2 23.0%

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

       6. associate-*r/ 23.0%

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

       7. distribute-rgt-in 98.3%

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

       8. associate-*l* 98.3%

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

       9. +-commutative 98.3%

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

   12. Simplified 98.3%

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

4. Final simplification 98.2%

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

Alternative 7: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

      1. *-commutative 100.0%

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

      2. flip-- 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 97.8%

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

      1. sub-neg 97.8%

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

      2. *-commutative 97.8%

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

      3. distribute-rgt-in 97.8%

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

      4. *-lft-identity 97.8%

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

      5. associate-*l/ 98.0%

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

      6. *-lft-identity 98.0%

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

      7. metadata-eval 98.0%

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

   10. Simplified 98.0%

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

   11. Taylor expanded in v around 0 98.0%

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

   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/ 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around inf 98.4%

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

      1. unpow2 98.4%

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

      2. associate-*l/ 98.3%

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

      3. neg-mul-1 98.3%

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

      4. distribute-rgt-neg-out 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in v around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unpow2 98.4%

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

      3. associate-/l* 98.4%

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

      4. distribute-neg-frac 98.4%

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

      5. distribute-rgt-neg-in 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in m around 0 23.0%

       \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   11. Step-by-step derivation

       1. +-commutative 23.0%

          \[\leadsto \color{blue}{\frac{{m}^{3}}{v} + -1 \cdot \frac{{m}^{2}}{v}} \]

       2. unpow3 23.0%

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

       3. associate-*r/ 23.0%

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

       4. associate-*r* 23.0%

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

       5. unpow2 23.0%

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

       6. associate-*r/ 23.0%

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

       7. distribute-rgt-in 98.3%

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

       8. associate-*l* 98.3%

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

       9. +-commutative 98.3%

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

   12. Simplified 98.3%

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

   13. Taylor expanded in v around 0 98.4%

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

4. Final simplification 98.2%

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

Alternative 8: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m * Float64(Float64(m * Float64(m + -1.0)) / 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);
	else
		tmp = m * ((m * (m + -1.0)) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m * N[(m + -1.0), $MachinePrecision]), $MachinePrecision] / v), $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. Taylor expanded in m around 0 98.1%

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

   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/ 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around inf 98.4%

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

      1. unpow2 98.4%

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

      2. associate-*l/ 98.3%

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

      3. neg-mul-1 98.3%

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

      4. distribute-rgt-neg-out 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in v around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unpow2 98.4%

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

      3. associate-/l* 98.4%

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

      4. distribute-neg-frac 98.4%

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

      5. distribute-rgt-neg-in 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in m around 0 23.0%

       \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   11. Step-by-step derivation

       1. +-commutative 23.0%

          \[\leadsto \color{blue}{\frac{{m}^{3}}{v} + -1 \cdot \frac{{m}^{2}}{v}} \]

       2. unpow3 23.0%

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

       3. associate-*r/ 23.0%

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

       4. associate-*r* 23.0%

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

       5. unpow2 23.0%

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

       6. associate-*r/ 23.0%

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

       7. distribute-rgt-in 98.3%

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

       8. associate-*l* 98.3%

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

       9. +-commutative 98.3%

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

   12. Simplified 98.3%

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

   13. Taylor expanded in v around 0 98.4%

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

4. Final simplification 98.2%

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

Alternative 9: 73.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.22 \cdot 10^{-117}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.22e-117) -1.0 (if (<= m 2.2) (/ m v) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.22e-117) {
		tmp = -1.0;
	} else if (m <= 2.2) {
		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 <= 1.22d-117) then
        tmp = -1.0d0
    else if (m <= 2.2d0) 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 <= 1.22e-117) {
		tmp = -1.0;
	} else if (m <= 2.2) {
		tmp = m / v;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.22e-117:
		tmp = -1.0
	elif m <= 2.2:
		tmp = m / v
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.22e-117)
		tmp = -1.0;
	elseif (m <= 2.2)
		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 <= 1.22e-117)
		tmp = -1.0;
	elseif (m <= 2.2)
		tmp = m / v;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.22e-117], -1.0, If[LessEqual[m, 2.2], N[(m / v), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.21999999999999997e-117

   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 69.2%

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

   if 1.21999999999999997e-117 < m < 2.2000000000000002

   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 94.5%

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

      1. sub-neg 94.5%

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

      2. distribute-rgt-in 94.5%

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

      3. *-un-lft-identity 94.5%

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

      4. sub-neg 94.5%

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

      5. metadata-eval 94.5%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 94.1%

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

      8. sqr-neg 94.1%

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

      9. sqrt-prod 94.1%

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

      10. add-sqr-sqrt 94.1%

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

      11. sub-neg 94.1%

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

      12. metadata-eval 94.1%

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

   4. Applied egg-rr 94.1%

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

      1. distribute-rgt1-in 94.1%

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

      2. +-commutative 94.1%

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

      3. +-commutative 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in v around 0 80.0%

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

      1. associate-*r/ 79.7%

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

   9. Simplified 79.7%

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

   10. Taylor expanded in m around 0 80.2%

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

   if 2.2000000000000002 < 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-rgt-in 0.1%

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

      3. *-un-lft-identity 0.1%

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

      4. sub-neg 0.1%

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

      5. metadata-eval 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 77.6%

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

      8. sqr-neg 77.6%

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

      9. sqrt-prod 77.6%

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

      10. add-sqr-sqrt 77.6%

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

      11. sub-neg 77.6%

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

      12. metadata-eval 77.6%

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

   4. Applied egg-rr 77.6%

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

      1. distribute-rgt1-in 77.6%

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

      2. +-commutative 77.6%

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

      3. +-commutative 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in m around inf 77.6%

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

      1. unpow2 77.6%

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

      2. associate-*r/ 77.6%

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

   9. Simplified 77.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.22 \cdot 10^{-117}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 10: 87.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. flip-- 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 97.8%

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

      1. sub-neg 97.8%

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

      2. *-commutative 97.8%

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

      3. distribute-rgt-in 97.8%

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

      4. *-lft-identity 97.8%

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

      5. associate-*l/ 98.0%

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

      6. *-lft-identity 98.0%

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

      7. metadata-eval 98.0%

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

   10. Simplified 98.0%

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

   11. Taylor expanded in v around 0 98.0%

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

   if 2.39999999999999991 < 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-rgt-in 0.1%

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

      3. *-un-lft-identity 0.1%

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

      4. sub-neg 0.1%

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

      5. metadata-eval 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 77.6%

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

      8. sqr-neg 77.6%

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

      9. sqrt-prod 77.6%

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

      10. add-sqr-sqrt 77.6%

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

      11. sub-neg 77.6%

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

      12. metadata-eval 77.6%

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

   4. Applied egg-rr 77.6%

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

      1. distribute-rgt1-in 77.6%

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

      2. +-commutative 77.6%

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

      3. +-commutative 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in v around 0 77.6%

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

      1. associate-*r/ 77.6%

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

   9. Simplified 77.6%

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

4. Final simplification 87.3%

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

Alternative 11: 87.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.2) (+ (/ m v) -1.0) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.2) {
		tmp = (m / v) + -1.0;
	} 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 <= 2.2d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.2) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.2:
		tmp = (m / v) + -1.0
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.2)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.2)
		tmp = (m / v) + -1.0;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.2], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.2000000000000002

   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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. flip-- 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 97.8%

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

      1. sub-neg 97.8%

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

      2. *-commutative 97.8%

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

      3. distribute-rgt-in 97.8%

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

      4. *-lft-identity 97.8%

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

      5. associate-*l/ 98.0%

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

      6. *-lft-identity 98.0%

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

      7. metadata-eval 98.0%

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

   10. Simplified 98.0%

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

   11. Taylor expanded in v around 0 98.0%

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

   if 2.2000000000000002 < 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-rgt-in 0.1%

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

      3. *-un-lft-identity 0.1%

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

      4. sub-neg 0.1%

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

      5. metadata-eval 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 77.6%

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

      8. sqr-neg 77.6%

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

      9. sqrt-prod 77.6%

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

      10. add-sqr-sqrt 77.6%

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

      11. sub-neg 77.6%

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

      12. metadata-eval 77.6%

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

   4. Applied egg-rr 77.6%

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

      1. distribute-rgt1-in 77.6%

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

      2. +-commutative 77.6%

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

      3. +-commutative 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in m around inf 77.6%

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

      1. unpow2 77.6%

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

      2. associate-*r/ 77.6%

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

   9. Simplified 77.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 12: 60.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.22 \cdot 10^{-117}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.22e-117) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.22e-117) {
		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 <= 1.22d-117) 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 <= 1.22e-117) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.22e-117:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.22e-117)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.22e-117)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.22e-117], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.21999999999999997e-117

   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 69.2%

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

   if 1.21999999999999997e-117 < 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 22.6%

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

      1. sub-neg 22.6%

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

      2. distribute-rgt-in 22.6%

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

      3. *-un-lft-identity 22.6%

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

      4. sub-neg 22.6%

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

      5. metadata-eval 22.6%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 81.5%

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

      8. sqr-neg 81.5%

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

      9. sqrt-prod 81.5%

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

      10. add-sqr-sqrt 81.5%

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

      11. sub-neg 81.5%

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

      12. metadata-eval 81.5%

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

   4. Applied egg-rr 81.5%

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

      1. distribute-rgt1-in 81.5%

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

      2. +-commutative 81.5%

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

      3. +-commutative 81.5%

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

   6. Simplified 81.5%

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

   7. Taylor expanded in v around 0 78.2%

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

      1. associate-*r/ 78.1%

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

   9. Simplified 78.1%

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

   10. Taylor expanded in m around 0 60.6%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.22 \cdot 10^{-117}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 13: 27.0% 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%

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

   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 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 14: 24.6% 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%

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

   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 24.1%

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

5. Final simplification 24.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023215 
(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)))