b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 13

Speedup: 1.0×

• 41.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. m = 1.2378326724065385e-27
  2. v = 6.861462539781798e-229

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 88.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.68:\\ \;\;\;\;-1 + m \cdot \frac{1 + m \cdot -2}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{m}{v}\right) \cdot \left(m + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.68)
   (+ -1.0 (* m (/ (+ 1.0 (* m -2.0)) v)))
   (* (+ 1.0 (/ m v)) (+ m -1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.68) {
		tmp = -1.0 + (m * ((1.0 + (m * -2.0)) / v));
	} else {
		tmp = (1.0 + (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 <= 0.68d0) then
        tmp = (-1.0d0) + (m * ((1.0d0 + (m * (-2.0d0))) / v))
    else
        tmp = (1.0d0 + (m / v)) * (m + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.68) {
		tmp = -1.0 + (m * ((1.0 + (m * -2.0)) / v));
	} else {
		tmp = (1.0 + (m / v)) * (m + -1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.68:
		tmp = -1.0 + (m * ((1.0 + (m * -2.0)) / v))
	else:
		tmp = (1.0 + (m / v)) * (m + -1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.68)
		tmp = Float64(-1.0 + Float64(m * Float64(Float64(1.0 + Float64(m * -2.0)) / v)));
	else
		tmp = Float64(Float64(1.0 + Float64(m / v)) * Float64(m + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.68)
		tmp = -1.0 + (m * ((1.0 + (m * -2.0)) / v));
	else
		tmp = (1.0 + (m / v)) * (m + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.68], N[(-1.0 + N[(m * N[(N[(1.0 + N[(m * -2.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(m + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.680000000000000049

   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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 97.7%

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

   6. Taylor expanded in v around 0 97.8%

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

      1. associate-/l* 97.5%

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

      2. *-commutative 97.5%

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

   8. Simplified 97.5%

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

   if 0.680000000000000049 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-rgt-in 0.1%

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

      5. associate-/r/ 0.1%

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

      6. div-inv 0.1%

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

      7. clear-num 0.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 74.0%

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

      10. sqr-neg 74.0%

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

      11. sqrt-unprod 74.0%

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

      12. add-sqr-sqrt 74.0%

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

      13. neg-sub0 74.0%

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

      14. associate--r- 74.0%

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

      15. metadata-eval 74.0%

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

      16. neg-mul-1 74.0%

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

      17. neg-sub0 74.0%

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

      18. associate--r- 74.0%

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

      19. metadata-eval 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. associate-*r/ 74.0%

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

      2. associate-*l/ 74.0%

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

      3. distribute-lft1-in 74.0%

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

      4. +-commutative 74.0%

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

   7. Simplified 74.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.68:\\ \;\;\;\;-1 + m \cdot \frac{1 + m \cdot -2}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{m}{v}\right) \cdot \left(m + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.68:\\ \;\;\;\;-1 + \frac{m \cdot \left(1 + m \cdot -2\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{m}{v}\right) \cdot \left(m + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.68)
   (+ -1.0 (/ (* m (+ 1.0 (* m -2.0))) v))
   (* (+ 1.0 (/ m v)) (+ m -1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.68) {
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	} else {
		tmp = (1.0 + (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 <= 0.68d0) then
        tmp = (-1.0d0) + ((m * (1.0d0 + (m * (-2.0d0)))) / v)
    else
        tmp = (1.0d0 + (m / v)) * (m + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.68) {
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	} else {
		tmp = (1.0 + (m / v)) * (m + -1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.68:
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v)
	else:
		tmp = (1.0 + (m / v)) * (m + -1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.68)
		tmp = Float64(-1.0 + Float64(Float64(m * Float64(1.0 + Float64(m * -2.0))) / v));
	else
		tmp = Float64(Float64(1.0 + Float64(m / v)) * Float64(m + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.68)
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	else
		tmp = (1.0 + (m / v)) * (m + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.68], N[(-1.0 + N[(N[(m * N[(1.0 + N[(m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(m + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.680000000000000049

   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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 97.7%

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

   6. Taylor expanded in v around 0 97.8%

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

   if 0.680000000000000049 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-rgt-in 0.1%

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

      5. associate-/r/ 0.1%

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

      6. div-inv 0.1%

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

      7. clear-num 0.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 74.0%

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

      10. sqr-neg 74.0%

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

      11. sqrt-unprod 74.0%

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

      12. add-sqr-sqrt 74.0%

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

      13. neg-sub0 74.0%

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

      14. associate--r- 74.0%

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

      15. metadata-eval 74.0%

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

      16. neg-mul-1 74.0%

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

      17. neg-sub0 74.0%

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

      18. associate--r- 74.0%

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

      19. metadata-eval 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. associate-*r/ 74.0%

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

      2. associate-*l/ 74.0%

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

      3. distribute-lft1-in 74.0%

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

      4. +-commutative 74.0%

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

   7. Simplified 74.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.68:\\ \;\;\;\;-1 + \frac{m \cdot \left(1 + m \cdot -2\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{m}{v}\right) \cdot \left(m + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.62) {
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	} 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 <= 0.62d0) then
        tmp = (-1.0d0) + ((m * (1.0d0 + (m * (-2.0d0)))) / v)
    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 <= 0.62) {
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	} else {
		tmp = (1.0 - m) * (-1.0 - (m / (v / m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.62:
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v)
	else:
		tmp = (1.0 - m) * (-1.0 - (m / (v / m)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.62)
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	else
		tmp = (1.0 - m) * (-1.0 - (m / (v / m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.62], N[(-1.0 + N[(N[(m * N[(1.0 + N[(m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $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 < 0.619999999999999996

   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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 97.7%

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

   6. Taylor expanded in v around 0 97.8%

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

   if 0.619999999999999996 < m

   1. Initial program 99.9%

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

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in m around inf 95.7%

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

      1. associate-*r/ 95.7%

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

      2. neg-mul-1 95.7%

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

   9. Simplified 95.7%

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

4. Final simplification 96.7%

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

Alternative 5: 88.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (1.0 + (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 / v)));
	else
		tmp = Float64(Float64(1.0 + 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));
	else
		tmp = (1.0 + (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 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(m + -1.0), $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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 97.7%

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

   6. Taylor expanded in m around 0 96.1%

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

      1. distribute-lft-in 96.1%

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

      2. *-rgt-identity 96.1%

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

      3. associate-*r/ 96.4%

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

      4. *-rgt-identity 96.4%

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

   8. Simplified 96.4%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-rgt-in 0.1%

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

      5. associate-/r/ 0.1%

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

      6. div-inv 0.1%

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

      7. clear-num 0.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 74.0%

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

      10. sqr-neg 74.0%

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

      11. sqrt-unprod 74.0%

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

      12. add-sqr-sqrt 74.0%

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

      13. neg-sub0 74.0%

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

      14. associate--r- 74.0%

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

      15. metadata-eval 74.0%

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

      16. neg-mul-1 74.0%

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

      17. neg-sub0 74.0%

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

      18. associate--r- 74.0%

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

      19. metadata-eval 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. associate-*r/ 74.0%

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

      2. associate-*l/ 74.0%

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

      3. distribute-lft1-in 74.0%

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

      4. +-commutative 74.0%

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

   7. Simplified 74.0%

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

4. Final simplification 85.1%

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

Alternative 6: 88.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

   3. Taylor expanded in m around 0 96.5%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-rgt-in 0.1%

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

      5. associate-/r/ 0.1%

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

      6. div-inv 0.1%

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

      7. clear-num 0.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 74.0%

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

      10. sqr-neg 74.0%

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

      11. sqrt-unprod 74.0%

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

      12. add-sqr-sqrt 74.0%

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

      13. neg-sub0 74.0%

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

      14. associate--r- 74.0%

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

      15. metadata-eval 74.0%

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

      16. neg-mul-1 74.0%

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

      17. neg-sub0 74.0%

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

      18. associate--r- 74.0%

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

      19. metadata-eval 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. associate-*r/ 74.0%

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

      2. associate-*l/ 74.0%

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

      3. distribute-lft1-in 74.0%

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

      4. +-commutative 74.0%

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

   7. Simplified 74.0%

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

4. Final simplification 85.2%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

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(m * Float64(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[(m * N[(N[(1.0 - m), $MachinePrecision] / 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.7%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 9: 88.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 97.7%

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

   6. Taylor expanded in v around 0 97.8%

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

   7. Taylor expanded in m around 0 96.2%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-rgt-in 0.1%

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

      5. associate-/r/ 0.1%

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

      6. div-inv 0.1%

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

      7. clear-num 0.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 74.0%

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

      10. sqr-neg 74.0%

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

      11. sqrt-unprod 74.0%

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

      12. add-sqr-sqrt 74.0%

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

      13. neg-sub0 74.0%

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

      14. associate--r- 74.0%

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

      15. metadata-eval 74.0%

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

      16. neg-mul-1 74.0%

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

      17. neg-sub0 74.0%

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

      18. associate--r- 74.0%

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

      19. metadata-eval 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. associate-*r/ 74.0%

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

      2. associate-*l/ 74.0%

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

      3. distribute-lft1-in 74.0%

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

      4. +-commutative 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in v around 0 74.0%

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

      1. sub-neg 74.0%

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

      2. metadata-eval 74.0%

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

      3. associate-*r/ 74.0%

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

   10. Simplified 74.0%

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

4. Final simplification 85.0%

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

Alternative 10: 88.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m + -1.0), $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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 97.7%

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

   6. Taylor expanded in m around 0 96.1%

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

      1. distribute-lft-in 96.1%

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

      2. *-rgt-identity 96.1%

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

      3. associate-*r/ 96.4%

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

      4. *-rgt-identity 96.4%

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

   8. Simplified 96.4%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-rgt-in 0.1%

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

      5. associate-/r/ 0.1%

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

      6. div-inv 0.1%

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

      7. clear-num 0.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 74.0%

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

      10. sqr-neg 74.0%

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

      11. sqrt-unprod 74.0%

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

      12. add-sqr-sqrt 74.0%

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

      13. neg-sub0 74.0%

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

      14. associate--r- 74.0%

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

      15. metadata-eval 74.0%

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

      16. neg-mul-1 74.0%

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

      17. neg-sub0 74.0%

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

      18. associate--r- 74.0%

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

      19. metadata-eval 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. associate-*r/ 74.0%

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

      2. associate-*l/ 74.0%

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

      3. distribute-lft1-in 74.0%

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

      4. +-commutative 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in v around 0 74.0%

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

      1. sub-neg 74.0%

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

      2. metadata-eval 74.0%

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

      3. associate-*r/ 74.0%

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

   10. Simplified 74.0%

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

4. Final simplification 85.1%

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

Alternative 11: 76.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return -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 / v)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(-1.0 + N[(m / v), $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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 48.6%

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

6. Taylor expanded in v around 0 48.6%

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

7. Taylor expanded in m around 0 73.9%

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

8. Final simplification 73.9%

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

Alternative 12: 27.9% 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 27.1%

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

   1. neg-mul-1 27.1%

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

   2. sub-neg 27.1%

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

   3. +-commutative 27.1%

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

   4. distribute-neg-in 27.1%

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

   5. remove-double-neg 27.1%

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

   6. metadata-eval 27.1%

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

7. Simplified 27.1%

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

8. Final simplification 27.1%

   \[\leadsto m + -1 \]
9. Add Preprocessing

Alternative 13: 25.4% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 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%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 24.5%

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

6. Final simplification 24.5%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(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)))