Rosa's DopplerBench

Percentage Accurate: 72.0% → 97.9%

Time: 16.5s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

C

double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 * v) / ((t1 + u) * (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Python

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

C

double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 * v) / ((t1 + u) * (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Python

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (/ (- t1) (/ (+ t1 u) v)) (+ t1 u)))

C

double code(double u, double v, double t1) {
	return (-t1 / ((t1 + u) / v)) / (t1 + u);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 / ((t1 + u) / v)) / (t1 + u)
end function

Java

public static double code(double u, double v, double t1) {
	return (-t1 / ((t1 + u) / v)) / (t1 + u);
}

Python

def code(u, v, t1):
	return (-t1 / ((t1 + u) / v)) / (t1 + u)

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) / Float64(Float64(t1 + u) / v)) / Float64(t1 + u))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 / ((t1 + u) / v)) / (t1 + u);
end

Wolfram

code[u_, v_, t1_] := N[(N[((-t1) / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/r* 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

   2. associate-/l* 99.3%

      \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]

4. Final simplification 99.3%

   \[\leadsto \frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u} \]

Alternative 2: 89.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -1.24 \cdot 10^{+141}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -7.6 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{-212}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \left(\frac{u}{t1} + -1\right)}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -1.24e+141)
     (/ (- v) (+ t1 u))
     (if (<= t1 -7.6e-132)
       t_1
       (if (<= t1 5.6e-212)
         (/ (* t1 (/ v u)) (- u))
         (if (<= t1 1.35e+154) t_1 (/ (* v (+ (/ u t1) -1.0)) (+ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.24e+141) {
		tmp = -v / (t1 + u);
	} else if (t1 <= -7.6e-132) {
		tmp = t_1;
	} else if (t1 <= 5.6e-212) {
		tmp = (t1 * (v / u)) / -u;
	} else if (t1 <= 1.35e+154) {
		tmp = t_1;
	} else {
		tmp = (v * ((u / t1) + -1.0)) / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-1.24d+141)) then
        tmp = -v / (t1 + u)
    else if (t1 <= (-7.6d-132)) then
        tmp = t_1
    else if (t1 <= 5.6d-212) then
        tmp = (t1 * (v / u)) / -u
    else if (t1 <= 1.35d+154) then
        tmp = t_1
    else
        tmp = (v * ((u / t1) + (-1.0d0))) / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.24e+141) {
		tmp = -v / (t1 + u);
	} else if (t1 <= -7.6e-132) {
		tmp = t_1;
	} else if (t1 <= 5.6e-212) {
		tmp = (t1 * (v / u)) / -u;
	} else if (t1 <= 1.35e+154) {
		tmp = t_1;
	} else {
		tmp = (v * ((u / t1) + -1.0)) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -1.24e+141:
		tmp = -v / (t1 + u)
	elif t1 <= -7.6e-132:
		tmp = t_1
	elif t1 <= 5.6e-212:
		tmp = (t1 * (v / u)) / -u
	elif t1 <= 1.35e+154:
		tmp = t_1
	else:
		tmp = (v * ((u / t1) + -1.0)) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -1.24e+141)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= -7.6e-132)
		tmp = t_1;
	elseif (t1 <= 5.6e-212)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	elseif (t1 <= 1.35e+154)
		tmp = t_1;
	else
		tmp = Float64(Float64(v * Float64(Float64(u / t1) + -1.0)) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -1.24e+141)
		tmp = -v / (t1 + u);
	elseif (t1 <= -7.6e-132)
		tmp = t_1;
	elseif (t1 <= 5.6e-212)
		tmp = (t1 * (v / u)) / -u;
	elseif (t1 <= 1.35e+154)
		tmp = t_1;
	else
		tmp = (v * ((u / t1) + -1.0)) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.24e+141], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -7.6e-132], t$95$1, If[LessEqual[t1, 5.6e-212], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[t1, 1.35e+154], t$95$1, N[(N[(v * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.24e141

   1. Initial program 47.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 70.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      2. associate-/l* 100.0%

         \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 97.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   5. Step-by-step derivation

      1. neg-mul-1 97.2%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   6. Simplified 97.2%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -1.24e141 < t1 < -7.5999999999999994e-132 or 5.60000000000000027e-212 < t1 < 1.35000000000000003e154

   1. Initial program 85.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 91.4%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 91.4%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   if -7.5999999999999994e-132 < t1 < 5.60000000000000027e-212

   1. Initial program 74.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 74.2%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 74.2%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   4. Taylor expanded in t1 around 0 74.2%

      \[\leadsto v \cdot \color{blue}{\left(-1 \cdot \frac{t1}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 74.2%

         \[\leadsto v \cdot \color{blue}{\frac{-1 \cdot t1}{{u}^{2}}} \]

      2. neg-mul-1 74.2%

         \[\leadsto v \cdot \frac{\color{blue}{-t1}}{{u}^{2}} \]

      3. unpow2 74.2%

         \[\leadsto v \cdot \frac{-t1}{\color{blue}{u \cdot u}} \]

   6. Simplified 74.2%

      \[\leadsto v \cdot \color{blue}{\frac{-t1}{u \cdot u}} \]
   7. Step-by-step derivation

      1. add-log-exp 56.7%

         \[\leadsto \color{blue}{\log \left(e^{v \cdot \frac{-t1}{u \cdot u}}\right)} \]

      2. exp-prod 55.8%

         \[\leadsto \log \color{blue}{\left({\left(e^{v}\right)}^{\left(\frac{-t1}{u \cdot u}\right)}\right)} \]

      3. add-sqr-sqrt 47.7%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{u \cdot u}\right)}\right) \]

      4. sqrt-unprod 51.6%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{u \cdot u}\right)}\right) \]

      5. sqr-neg 51.6%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{u \cdot u}\right)}\right) \]

      6. sqrt-unprod 35.4%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{u \cdot u}\right)}\right) \]

      7. add-sqr-sqrt 49.3%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{t1}}{u \cdot u}\right)}\right) \]

   8. Applied egg-rr 49.3%

      \[\leadsto \color{blue}{\log \left({\left(e^{v}\right)}^{\left(\frac{t1}{u \cdot u}\right)}\right)} \]
   9. Step-by-step derivation

      1. log-pow 27.4%

         \[\leadsto \color{blue}{\frac{t1}{u \cdot u} \cdot \log \left(e^{v}\right)} \]

      2. rem-log-exp 49.0%

         \[\leadsto \frac{t1}{u \cdot u} \cdot \color{blue}{v} \]

   10. Simplified 49.0%

       \[\leadsto \color{blue}{\frac{t1}{u \cdot u} \cdot v} \]
   11. Step-by-step derivation

       1. associate-*l/ 48.9%

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{u \cdot u}} \]

       2. associate-/l* 48.9%

          \[\leadsto \color{blue}{\frac{t1}{\frac{u \cdot u}{v}}} \]

       3. add-sqr-sqrt 17.5%

          \[\leadsto \frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{\frac{u \cdot u}{v}} \]

       4. sqrt-unprod 53.8%

          \[\leadsto \frac{\color{blue}{\sqrt{t1 \cdot t1}}}{\frac{u \cdot u}{v}} \]

       5. sqr-neg 53.8%

          \[\leadsto \frac{\sqrt{\color{blue}{\left(-t1\right) \cdot \left(-t1\right)}}}{\frac{u \cdot u}{v}} \]

       6. sqrt-unprod 47.8%

          \[\leadsto \frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{\frac{u \cdot u}{v}} \]

       7. add-sqr-sqrt 75.1%

          \[\leadsto \frac{\color{blue}{-t1}}{\frac{u \cdot u}{v}} \]

       8. associate-/l* 74.6%

          \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{u \cdot u}} \]

       9. associate-/r* 82.1%

          \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{u}}{u}} \]

       10. frac-2neg 82.1%

           \[\leadsto \color{blue}{\frac{-\frac{\left(-t1\right) \cdot v}{u}}{-u}} \]

   12. Applied egg-rr 91.6%

       \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{u}}{-u}} \]

   if 1.35000000000000003e154 < t1

   1. Initial program 33.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.2%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      2. associate-/l* 99.1%

         \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 74.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot v + \frac{u \cdot v}{t1}}}{t1 + u} \]
   5. Step-by-step derivation

      1. neg-mul-1 74.9%

         \[\leadsto \frac{\color{blue}{\left(-v\right)} + \frac{u \cdot v}{t1}}{t1 + u} \]

      2. +-commutative 74.9%

         \[\leadsto \frac{\color{blue}{\frac{u \cdot v}{t1} + \left(-v\right)}}{t1 + u} \]

      3. unsub-neg 74.9%

         \[\leadsto \frac{\color{blue}{\frac{u \cdot v}{t1} - v}}{t1 + u} \]

      4. associate-/l* 84.1%

         \[\leadsto \frac{\color{blue}{\frac{u}{\frac{t1}{v}}} - v}{t1 + u} \]

   6. Simplified 84.1%

      \[\leadsto \frac{\color{blue}{\frac{u}{\frac{t1}{v}} - v}}{t1 + u} \]

   7. Taylor expanded in v around 0 84.1%

      \[\leadsto \color{blue}{\frac{v \cdot \left(\frac{u}{t1} - 1\right)}{t1 + u}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.24 \cdot 10^{+141}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -7.6 \cdot 10^{-132}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{-212}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \left(\frac{u}{t1} + -1\right)}{t1 + u}\\ \end{array} \]

Alternative 3: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := \frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{if}\;t1 \leq -3.85 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.05 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -4.3 \cdot 10^{-109}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 2.3 \cdot 10^{-54}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))) (t_2 (* (/ v u) (/ t1 (- u)))))
   (if (<= t1 -3.85e+55)
     t_1
     (if (<= t1 -1.05e-37)
       t_2
       (if (<= t1 -4.3e-109) (/ v (- u t1)) (if (<= t1 2.3e-54) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = (v / u) * (t1 / -u);
	double tmp;
	if (t1 <= -3.85e+55) {
		tmp = t_1;
	} else if (t1 <= -1.05e-37) {
		tmp = t_2;
	} else if (t1 <= -4.3e-109) {
		tmp = v / (u - t1);
	} else if (t1 <= 2.3e-54) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    t_2 = (v / u) * (t1 / -u)
    if (t1 <= (-3.85d+55)) then
        tmp = t_1
    else if (t1 <= (-1.05d-37)) then
        tmp = t_2
    else if (t1 <= (-4.3d-109)) then
        tmp = v / (u - t1)
    else if (t1 <= 2.3d-54) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = (v / u) * (t1 / -u);
	double tmp;
	if (t1 <= -3.85e+55) {
		tmp = t_1;
	} else if (t1 <= -1.05e-37) {
		tmp = t_2;
	} else if (t1 <= -4.3e-109) {
		tmp = v / (u - t1);
	} else if (t1 <= 2.3e-54) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	t_2 = (v / u) * (t1 / -u)
	tmp = 0
	if t1 <= -3.85e+55:
		tmp = t_1
	elif t1 <= -1.05e-37:
		tmp = t_2
	elif t1 <= -4.3e-109:
		tmp = v / (u - t1)
	elif t1 <= 2.3e-54:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	t_2 = Float64(Float64(v / u) * Float64(t1 / Float64(-u)))
	tmp = 0.0
	if (t1 <= -3.85e+55)
		tmp = t_1;
	elseif (t1 <= -1.05e-37)
		tmp = t_2;
	elseif (t1 <= -4.3e-109)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= 2.3e-54)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	t_2 = (v / u) * (t1 / -u);
	tmp = 0.0;
	if (t1 <= -3.85e+55)
		tmp = t_1;
	elseif (t1 <= -1.05e-37)
		tmp = t_2;
	elseif (t1 <= -4.3e-109)
		tmp = v / (u - t1);
	elseif (t1 <= 2.3e-54)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.85e+55], t$95$1, If[LessEqual[t1, -1.05e-37], t$95$2, If[LessEqual[t1, -4.3e-109], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.3e-54], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.84999999999999984e55 or 2.2999999999999999e-54 < t1

   1. Initial program 60.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 75.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      2. associate-/l* 99.6%

         \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 85.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   5. Step-by-step derivation

      1. neg-mul-1 85.0%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   6. Simplified 85.0%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -3.84999999999999984e55 < t1 < -1.05e-37 or -4.2999999999999997e-109 < t1 < 2.2999999999999999e-54

   1. Initial program 80.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 81.9%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      2. neg-mul-1 81.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. associate-*r/ 91.5%

         \[\leadsto \frac{-1 \cdot t1}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{v}}} \]

      4. times-frac 98.7%

         \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \frac{t1}{\frac{t1 + u}{v}}} \]

      5. div-inv 98.6%

         \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{\left(t1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

      6. clear-num 98.7%

         \[\leadsto \frac{-1}{t1 + u} \cdot \left(t1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \]

   3. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)} \]

   4. Taylor expanded in t1 around 0 72.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   5. Step-by-step derivation

      1. metadata-eval 72.1%

         \[\leadsto \color{blue}{\frac{1}{-1}} \cdot \frac{t1 \cdot v}{{u}^{2}} \]

      2. *-commutative 72.1%

         \[\leadsto \frac{1}{-1} \cdot \frac{\color{blue}{v \cdot t1}}{{u}^{2}} \]

      3. unpow2 72.1%

         \[\leadsto \frac{1}{-1} \cdot \frac{v \cdot t1}{\color{blue}{u \cdot u}} \]

      4. associate-/r* 79.1%

         \[\leadsto \frac{1}{-1} \cdot \color{blue}{\frac{\frac{v \cdot t1}{u}}{u}} \]

      5. *-commutative 79.1%

         \[\leadsto \frac{1}{-1} \cdot \frac{\frac{\color{blue}{t1 \cdot v}}{u}}{u} \]

      6. associate-*l/ 85.5%

         \[\leadsto \frac{1}{-1} \cdot \frac{\color{blue}{\frac{t1}{u} \cdot v}}{u} \]

      7. times-frac 85.5%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{t1}{u} \cdot v\right)}{-1 \cdot u}} \]

      8. *-lft-identity 85.5%

         \[\leadsto \frac{\color{blue}{\frac{t1}{u} \cdot v}}{-1 \cdot u} \]

      9. associate-*l/ 79.1%

         \[\leadsto \frac{\color{blue}{\frac{t1 \cdot v}{u}}}{-1 \cdot u} \]

      10. *-commutative 79.1%

          \[\leadsto \frac{\frac{\color{blue}{v \cdot t1}}{u}}{-1 \cdot u} \]

      11. neg-mul-1 79.1%

          \[\leadsto \frac{\frac{v \cdot t1}{u}}{\color{blue}{-u}} \]

      12. associate-/r* 72.1%

          \[\leadsto \color{blue}{\frac{v \cdot t1}{u \cdot \left(-u\right)}} \]

      13. times-frac 84.5%

          \[\leadsto \color{blue}{\frac{v}{u} \cdot \frac{t1}{-u}} \]

   6. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{v}{u} \cdot \frac{t1}{-u}} \]

   if -1.05e-37 < t1 < -4.2999999999999997e-109

   1. Initial program 92.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 92.8%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      2. neg-mul-1 92.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. associate-*r/ 93.1%

         \[\leadsto \frac{-1 \cdot t1}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{v}}} \]

      4. times-frac 99.4%

         \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \frac{t1}{\frac{t1 + u}{v}}} \]

      5. div-inv 99.4%

         \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{\left(t1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

      6. clear-num 99.6%

         \[\leadsto \frac{-1}{t1 + u} \cdot \left(t1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \]

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)} \]

   4. Taylor expanded in t1 around inf 82.4%

      \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{v} \]
   5. Step-by-step derivation

      1. *-commutative 82.4%

         \[\leadsto \color{blue}{v \cdot \frac{-1}{t1 + u}} \]

      2. frac-2neg 82.4%

         \[\leadsto v \cdot \color{blue}{\frac{--1}{-\left(t1 + u\right)}} \]

      3. metadata-eval 82.4%

         \[\leadsto v \cdot \frac{\color{blue}{1}}{-\left(t1 + u\right)} \]

      4. un-div-inv 82.6%

         \[\leadsto \color{blue}{\frac{v}{-\left(t1 + u\right)}} \]

      5. +-commutative 82.6%

         \[\leadsto \frac{v}{-\color{blue}{\left(u + t1\right)}} \]

      6. distribute-neg-in 82.6%

         \[\leadsto \frac{v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      7. add-sqr-sqrt 53.8%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)} \]

      8. sqrt-unprod 84.2%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \left(-t1\right)} \]

      9. sqr-neg 84.2%

         \[\leadsto \frac{v}{\sqrt{\color{blue}{u \cdot u}} + \left(-t1\right)} \]

      10. sqrt-unprod 30.4%

          \[\leadsto \frac{v}{\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \left(-t1\right)} \]

      11. add-sqr-sqrt 83.9%

          \[\leadsto \frac{v}{\color{blue}{u} + \left(-t1\right)} \]

   6. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{\frac{v}{u + \left(-t1\right)}} \]
   7. Step-by-step derivation

      1. unsub-neg 83.9%

         \[\leadsto \frac{v}{\color{blue}{u - t1}} \]

   8. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{v}{u - t1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.85 \cdot 10^{+55}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.05 \cdot 10^{-37}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;t1 \leq -4.3 \cdot 10^{-109}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 2.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 4: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -7 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -5.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;t1 \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -7e+54)
     t_1
     (if (<= t1 -5.6e-49)
       (/ (- t1) (/ u (/ v u)))
       (if (<= t1 -5e-109)
         (/ v (- u t1))
         (if (<= t1 2.2e-54) (* (/ v u) (/ t1 (- u))) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -7e+54) {
		tmp = t_1;
	} else if (t1 <= -5.6e-49) {
		tmp = -t1 / (u / (v / u));
	} else if (t1 <= -5e-109) {
		tmp = v / (u - t1);
	} else if (t1 <= 2.2e-54) {
		tmp = (v / u) * (t1 / -u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    if (t1 <= (-7d+54)) then
        tmp = t_1
    else if (t1 <= (-5.6d-49)) then
        tmp = -t1 / (u / (v / u))
    else if (t1 <= (-5d-109)) then
        tmp = v / (u - t1)
    else if (t1 <= 2.2d-54) then
        tmp = (v / u) * (t1 / -u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -7e+54) {
		tmp = t_1;
	} else if (t1 <= -5.6e-49) {
		tmp = -t1 / (u / (v / u));
	} else if (t1 <= -5e-109) {
		tmp = v / (u - t1);
	} else if (t1 <= 2.2e-54) {
		tmp = (v / u) * (t1 / -u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -7e+54:
		tmp = t_1
	elif t1 <= -5.6e-49:
		tmp = -t1 / (u / (v / u))
	elif t1 <= -5e-109:
		tmp = v / (u - t1)
	elif t1 <= 2.2e-54:
		tmp = (v / u) * (t1 / -u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -7e+54)
		tmp = t_1;
	elseif (t1 <= -5.6e-49)
		tmp = Float64(Float64(-t1) / Float64(u / Float64(v / u)));
	elseif (t1 <= -5e-109)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= 2.2e-54)
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -7e+54)
		tmp = t_1;
	elseif (t1 <= -5.6e-49)
		tmp = -t1 / (u / (v / u));
	elseif (t1 <= -5e-109)
		tmp = v / (u - t1);
	elseif (t1 <= 2.2e-54)
		tmp = (v / u) * (t1 / -u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7e+54], t$95$1, If[LessEqual[t1, -5.6e-49], N[((-t1) / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -5e-109], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.2e-54], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -7.0000000000000002e54 or 2.2e-54 < t1

   1. Initial program 60.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 75.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      2. associate-/l* 99.6%

         \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 85.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   5. Step-by-step derivation

      1. neg-mul-1 85.0%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   6. Simplified 85.0%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -7.0000000000000002e54 < t1 < -5.59999999999999995e-49

   1. Initial program 78.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 94.3%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      2. associate-/l* 99.7%

         \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]
   4. Step-by-step derivation

      1. clear-num 99.6%

         \[\leadsto \frac{\frac{-t1}{\color{blue}{\frac{1}{\frac{v}{t1 + u}}}}}{t1 + u} \]

      2. inv-pow 99.6%

         \[\leadsto \frac{\frac{-t1}{\color{blue}{{\left(\frac{v}{t1 + u}\right)}^{-1}}}}{t1 + u} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \frac{\frac{-t1}{\color{blue}{{\left(\frac{v}{t1 + u}\right)}^{-1}}}}{t1 + u} \]
   6. Step-by-step derivation

      1. unpow-1 99.6%

         \[\leadsto \frac{\frac{-t1}{\color{blue}{\frac{1}{\frac{v}{t1 + u}}}}}{t1 + u} \]

   7. Simplified 99.6%

      \[\leadsto \frac{\frac{-t1}{\color{blue}{\frac{1}{\frac{v}{t1 + u}}}}}{t1 + u} \]

   8. Taylor expanded in t1 around 0 52.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 52.1%

         \[\leadsto \color{blue}{-\frac{t1 \cdot v}{{u}^{2}}} \]

      2. unpow2 52.1%

         \[\leadsto -\frac{t1 \cdot v}{\color{blue}{u \cdot u}} \]

      3. times-frac 73.1%

         \[\leadsto -\color{blue}{\frac{t1}{u} \cdot \frac{v}{u}} \]

      4. associate-/r/ 73.3%

         \[\leadsto -\color{blue}{\frac{t1}{\frac{u}{\frac{v}{u}}}} \]

      5. distribute-neg-frac 73.3%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{u}{\frac{v}{u}}}} \]

   10. Simplified 73.3%

       \[\leadsto \color{blue}{\frac{-t1}{\frac{u}{\frac{v}{u}}}} \]

   if -5.59999999999999995e-49 < t1 < -5.0000000000000002e-109

   1. Initial program 92.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 92.8%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      2. neg-mul-1 92.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. associate-*r/ 93.1%

         \[\leadsto \frac{-1 \cdot t1}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{v}}} \]

      4. times-frac 99.4%

         \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \frac{t1}{\frac{t1 + u}{v}}} \]

      5. div-inv 99.4%

         \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{\left(t1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

      6. clear-num 99.6%

         \[\leadsto \frac{-1}{t1 + u} \cdot \left(t1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \]

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)} \]

   4. Taylor expanded in t1 around inf 82.4%

      \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{v} \]
   5. Step-by-step derivation

      1. *-commutative 82.4%

         \[\leadsto \color{blue}{v \cdot \frac{-1}{t1 + u}} \]

      2. frac-2neg 82.4%

         \[\leadsto v \cdot \color{blue}{\frac{--1}{-\left(t1 + u\right)}} \]

      3. metadata-eval 82.4%

         \[\leadsto v \cdot \frac{\color{blue}{1}}{-\left(t1 + u\right)} \]

      4. un-div-inv 82.6%

         \[\leadsto \color{blue}{\frac{v}{-\left(t1 + u\right)}} \]

      5. +-commutative 82.6%

         \[\leadsto \frac{v}{-\color{blue}{\left(u + t1\right)}} \]

      6. distribute-neg-in 82.6%

         \[\leadsto \frac{v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      7. add-sqr-sqrt 53.8%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)} \]

      8. sqrt-unprod 84.2%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \left(-t1\right)} \]

      9. sqr-neg 84.2%

         \[\leadsto \frac{v}{\sqrt{\color{blue}{u \cdot u}} + \left(-t1\right)} \]

      10. sqrt-unprod 30.4%

          \[\leadsto \frac{v}{\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \left(-t1\right)} \]

      11. add-sqr-sqrt 83.9%

          \[\leadsto \frac{v}{\color{blue}{u} + \left(-t1\right)} \]

   6. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{\frac{v}{u + \left(-t1\right)}} \]
   7. Step-by-step derivation

      1. unsub-neg 83.9%

         \[\leadsto \frac{v}{\color{blue}{u - t1}} \]

   8. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{v}{u - t1}} \]

   if -5.0000000000000002e-109 < t1 < 2.2e-54

   1. Initial program 80.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.7%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      2. neg-mul-1 83.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. associate-*r/ 90.9%

         \[\leadsto \frac{-1 \cdot t1}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{v}}} \]

      4. times-frac 98.5%

         \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \frac{t1}{\frac{t1 + u}{v}}} \]

      5. div-inv 98.4%

         \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{\left(t1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

      6. clear-num 98.5%

         \[\leadsto \frac{-1}{t1 + u} \cdot \left(t1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \]

   3. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)} \]

   4. Taylor expanded in t1 around 0 76.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   5. Step-by-step derivation

      1. metadata-eval 76.7%

         \[\leadsto \color{blue}{\frac{1}{-1}} \cdot \frac{t1 \cdot v}{{u}^{2}} \]

      2. *-commutative 76.7%

         \[\leadsto \frac{1}{-1} \cdot \frac{\color{blue}{v \cdot t1}}{{u}^{2}} \]

      3. unpow2 76.7%

         \[\leadsto \frac{1}{-1} \cdot \frac{v \cdot t1}{\color{blue}{u \cdot u}} \]

      4. associate-/r* 81.6%

         \[\leadsto \frac{1}{-1} \cdot \color{blue}{\frac{\frac{v \cdot t1}{u}}{u}} \]

      5. *-commutative 81.6%

         \[\leadsto \frac{1}{-1} \cdot \frac{\frac{\color{blue}{t1 \cdot v}}{u}}{u} \]

      6. associate-*l/ 88.3%

         \[\leadsto \frac{1}{-1} \cdot \frac{\color{blue}{\frac{t1}{u} \cdot v}}{u} \]

      7. times-frac 88.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{t1}{u} \cdot v\right)}{-1 \cdot u}} \]

      8. *-lft-identity 88.3%

         \[\leadsto \frac{\color{blue}{\frac{t1}{u} \cdot v}}{-1 \cdot u} \]

      9. associate-*l/ 81.6%

         \[\leadsto \frac{\color{blue}{\frac{t1 \cdot v}{u}}}{-1 \cdot u} \]

      10. *-commutative 81.6%

          \[\leadsto \frac{\frac{\color{blue}{v \cdot t1}}{u}}{-1 \cdot u} \]

      11. neg-mul-1 81.6%

          \[\leadsto \frac{\frac{v \cdot t1}{u}}{\color{blue}{-u}} \]

      12. associate-/r* 76.7%

          \[\leadsto \color{blue}{\frac{v \cdot t1}{u \cdot \left(-u\right)}} \]

      13. times-frac 87.0%

          \[\leadsto \color{blue}{\frac{v}{u} \cdot \frac{t1}{-u}} \]

   6. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{v}{u} \cdot \frac{t1}{-u}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+54}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -5.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;t1 \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 5: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.25 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -3.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;t1 \leq -2.05 \cdot 10^{-109}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 2.35 \cdot 10^{-48}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -3.25e+56)
     t_1
     (if (<= t1 -3.7e-41)
       (/ (- t1) (/ u (/ v u)))
       (if (<= t1 -2.05e-109)
         (/ v (- u t1))
         (if (<= t1 2.35e-48) (/ (* t1 (/ v u)) (- u)) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -3.25e+56) {
		tmp = t_1;
	} else if (t1 <= -3.7e-41) {
		tmp = -t1 / (u / (v / u));
	} else if (t1 <= -2.05e-109) {
		tmp = v / (u - t1);
	} else if (t1 <= 2.35e-48) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    if (t1 <= (-3.25d+56)) then
        tmp = t_1
    else if (t1 <= (-3.7d-41)) then
        tmp = -t1 / (u / (v / u))
    else if (t1 <= (-2.05d-109)) then
        tmp = v / (u - t1)
    else if (t1 <= 2.35d-48) then
        tmp = (t1 * (v / u)) / -u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -3.25e+56) {
		tmp = t_1;
	} else if (t1 <= -3.7e-41) {
		tmp = -t1 / (u / (v / u));
	} else if (t1 <= -2.05e-109) {
		tmp = v / (u - t1);
	} else if (t1 <= 2.35e-48) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -3.25e+56:
		tmp = t_1
	elif t1 <= -3.7e-41:
		tmp = -t1 / (u / (v / u))
	elif t1 <= -2.05e-109:
		tmp = v / (u - t1)
	elif t1 <= 2.35e-48:
		tmp = (t1 * (v / u)) / -u
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -3.25e+56)
		tmp = t_1;
	elseif (t1 <= -3.7e-41)
		tmp = Float64(Float64(-t1) / Float64(u / Float64(v / u)));
	elseif (t1 <= -2.05e-109)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= 2.35e-48)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -3.25e+56)
		tmp = t_1;
	elseif (t1 <= -3.7e-41)
		tmp = -t1 / (u / (v / u));
	elseif (t1 <= -2.05e-109)
		tmp = v / (u - t1);
	elseif (t1 <= 2.35e-48)
		tmp = (t1 * (v / u)) / -u;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.25e+56], t$95$1, If[LessEqual[t1, -3.7e-41], N[((-t1) / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -2.05e-109], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.35e-48], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -3.25e56 or 2.3499999999999999e-48 < t1

   1. Initial program 60.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 75.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      2. associate-/l* 99.6%

         \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 85.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   5. Step-by-step derivation

      1. neg-mul-1 85.0%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   6. Simplified 85.0%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -3.25e56 < t1 < -3.7000000000000002e-41

   1. Initial program 78.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 94.3%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      2. associate-/l* 99.7%

         \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]
   4. Step-by-step derivation

      1. clear-num 99.6%

         \[\leadsto \frac{\frac{-t1}{\color{blue}{\frac{1}{\frac{v}{t1 + u}}}}}{t1 + u} \]

      2. inv-pow 99.6%

         \[\leadsto \frac{\frac{-t1}{\color{blue}{{\left(\frac{v}{t1 + u}\right)}^{-1}}}}{t1 + u} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \frac{\frac{-t1}{\color{blue}{{\left(\frac{v}{t1 + u}\right)}^{-1}}}}{t1 + u} \]
   6. Step-by-step derivation

      1. unpow-1 99.6%

         \[\leadsto \frac{\frac{-t1}{\color{blue}{\frac{1}{\frac{v}{t1 + u}}}}}{t1 + u} \]

   7. Simplified 99.6%

      \[\leadsto \frac{\frac{-t1}{\color{blue}{\frac{1}{\frac{v}{t1 + u}}}}}{t1 + u} \]

   8. Taylor expanded in t1 around 0 52.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 52.1%

         \[\leadsto \color{blue}{-\frac{t1 \cdot v}{{u}^{2}}} \]

      2. unpow2 52.1%

         \[\leadsto -\frac{t1 \cdot v}{\color{blue}{u \cdot u}} \]

      3. times-frac 73.1%

         \[\leadsto -\color{blue}{\frac{t1}{u} \cdot \frac{v}{u}} \]

      4. associate-/r/ 73.3%

         \[\leadsto -\color{blue}{\frac{t1}{\frac{u}{\frac{v}{u}}}} \]

      5. distribute-neg-frac 73.3%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{u}{\frac{v}{u}}}} \]

   10. Simplified 73.3%

       \[\leadsto \color{blue}{\frac{-t1}{\frac{u}{\frac{v}{u}}}} \]

   if -3.7000000000000002e-41 < t1 < -2.0500000000000001e-109

   1. Initial program 92.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 92.8%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      2. neg-mul-1 92.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. associate-*r/ 93.1%

         \[\leadsto \frac{-1 \cdot t1}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{v}}} \]

      4. times-frac 99.4%

         \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \frac{t1}{\frac{t1 + u}{v}}} \]

      5. div-inv 99.4%

         \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{\left(t1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

      6. clear-num 99.6%

         \[\leadsto \frac{-1}{t1 + u} \cdot \left(t1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \]

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)} \]

   4. Taylor expanded in t1 around inf 82.4%

      \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{v} \]
   5. Step-by-step derivation

      1. *-commutative 82.4%

         \[\leadsto \color{blue}{v \cdot \frac{-1}{t1 + u}} \]

      2. frac-2neg 82.4%

         \[\leadsto v \cdot \color{blue}{\frac{--1}{-\left(t1 + u\right)}} \]

      3. metadata-eval 82.4%

         \[\leadsto v \cdot \frac{\color{blue}{1}}{-\left(t1 + u\right)} \]

      4. un-div-inv 82.6%

         \[\leadsto \color{blue}{\frac{v}{-\left(t1 + u\right)}} \]

      5. +-commutative 82.6%

         \[\leadsto \frac{v}{-\color{blue}{\left(u + t1\right)}} \]

      6. distribute-neg-in 82.6%

         \[\leadsto \frac{v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      7. add-sqr-sqrt 53.8%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)} \]

      8. sqrt-unprod 84.2%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \left(-t1\right)} \]

      9. sqr-neg 84.2%

         \[\leadsto \frac{v}{\sqrt{\color{blue}{u \cdot u}} + \left(-t1\right)} \]

      10. sqrt-unprod 30.4%

          \[\leadsto \frac{v}{\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \left(-t1\right)} \]

      11. add-sqr-sqrt 83.9%

          \[\leadsto \frac{v}{\color{blue}{u} + \left(-t1\right)} \]

   6. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{\frac{v}{u + \left(-t1\right)}} \]
   7. Step-by-step derivation

      1. unsub-neg 83.9%

         \[\leadsto \frac{v}{\color{blue}{u - t1}} \]

   8. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{v}{u - t1}} \]

   if -2.0500000000000001e-109 < t1 < 2.3499999999999999e-48

   1. Initial program 80.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 79.6%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 79.6%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 79.6%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   4. Taylor expanded in t1 around 0 74.5%

      \[\leadsto v \cdot \color{blue}{\left(-1 \cdot \frac{t1}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 74.5%

         \[\leadsto v \cdot \color{blue}{\frac{-1 \cdot t1}{{u}^{2}}} \]

      2. neg-mul-1 74.5%

         \[\leadsto v \cdot \frac{\color{blue}{-t1}}{{u}^{2}} \]

      3. unpow2 74.5%

         \[\leadsto v \cdot \frac{-t1}{\color{blue}{u \cdot u}} \]

   6. Simplified 74.5%

      \[\leadsto v \cdot \color{blue}{\frac{-t1}{u \cdot u}} \]
   7. Step-by-step derivation

      1. add-log-exp 51.7%

         \[\leadsto \color{blue}{\log \left(e^{v \cdot \frac{-t1}{u \cdot u}}\right)} \]

      2. exp-prod 51.0%

         \[\leadsto \log \color{blue}{\left({\left(e^{v}\right)}^{\left(\frac{-t1}{u \cdot u}\right)}\right)} \]

      3. add-sqr-sqrt 39.0%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{u \cdot u}\right)}\right) \]

      4. sqrt-unprod 46.8%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{u \cdot u}\right)}\right) \]

      5. sqr-neg 46.8%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{u \cdot u}\right)}\right) \]

      6. sqrt-unprod 34.9%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{u \cdot u}\right)}\right) \]

      7. add-sqr-sqrt 45.2%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{t1}}{u \cdot u}\right)}\right) \]

   8. Applied egg-rr 45.2%

      \[\leadsto \color{blue}{\log \left({\left(e^{v}\right)}^{\left(\frac{t1}{u \cdot u}\right)}\right)} \]
   9. Step-by-step derivation

      1. log-pow 25.6%

         \[\leadsto \color{blue}{\frac{t1}{u \cdot u} \cdot \log \left(e^{v}\right)} \]

      2. rem-log-exp 45.0%

         \[\leadsto \frac{t1}{u \cdot u} \cdot \color{blue}{v} \]

   10. Simplified 45.0%

       \[\leadsto \color{blue}{\frac{t1}{u \cdot u} \cdot v} \]
   11. Step-by-step derivation

       1. associate-*l/ 44.9%

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{u \cdot u}} \]

       2. associate-/l* 44.9%

          \[\leadsto \color{blue}{\frac{t1}{\frac{u \cdot u}{v}}} \]

       3. add-sqr-sqrt 23.2%

          \[\leadsto \frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{\frac{u \cdot u}{v}} \]

       4. sqrt-unprod 50.6%

          \[\leadsto \frac{\color{blue}{\sqrt{t1 \cdot t1}}}{\frac{u \cdot u}{v}} \]

       5. sqr-neg 50.6%

          \[\leadsto \frac{\sqrt{\color{blue}{\left(-t1\right) \cdot \left(-t1\right)}}}{\frac{u \cdot u}{v}} \]

       6. sqrt-unprod 34.9%

          \[\leadsto \frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{\frac{u \cdot u}{v}} \]

       7. add-sqr-sqrt 78.7%

          \[\leadsto \frac{\color{blue}{-t1}}{\frac{u \cdot u}{v}} \]

       8. associate-/l* 76.7%

          \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{u \cdot u}} \]

       9. associate-/r* 81.6%

          \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{u}}{u}} \]

       10. frac-2neg 81.6%

           \[\leadsto \color{blue}{\frac{-\frac{\left(-t1\right) \cdot v}{u}}{-u}} \]

   12. Applied egg-rr 89.4%

       \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{u}}{-u}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.25 \cdot 10^{+56}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -3.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;t1 \leq -2.05 \cdot 10^{-109}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 2.35 \cdot 10^{-48}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 6: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.8 \cdot 10^{-72}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.2e-108)
   (/ v (- u t1))
   (if (<= t1 1.8e-72) (* (- v) (/ t1 (* u u))) (/ (- v) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e-108) {
		tmp = v / (u - t1);
	} else if (t1 <= 1.8e-72) {
		tmp = -v * (t1 / (u * u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.2d-108)) then
        tmp = v / (u - t1)
    else if (t1 <= 1.8d-72) then
        tmp = -v * (t1 / (u * u))
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e-108) {
		tmp = v / (u - t1);
	} else if (t1 <= 1.8e-72) {
		tmp = -v * (t1 / (u * u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.2e-108:
		tmp = v / (u - t1)
	elif t1 <= 1.8e-72:
		tmp = -v * (t1 / (u * u))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.2e-108)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= 1.8e-72)
		tmp = Float64(Float64(-v) * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.2e-108)
		tmp = v / (u - t1);
	elseif (t1 <= 1.8e-72)
		tmp = -v * (t1 / (u * u));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.2e-108], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.8e-72], N[((-v) * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.20000000000000009e-108

   1. Initial program 71.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 67.6%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      2. neg-mul-1 67.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. associate-*r/ 80.2%

         \[\leadsto \frac{-1 \cdot t1}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{v}}} \]

      4. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \frac{t1}{\frac{t1 + u}{v}}} \]

      5. div-inv 99.6%

         \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{\left(t1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

      6. clear-num 99.7%

         \[\leadsto \frac{-1}{t1 + u} \cdot \left(t1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)} \]

   4. Taylor expanded in t1 around inf 81.1%

      \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{v} \]
   5. Step-by-step derivation

      1. *-commutative 81.1%

         \[\leadsto \color{blue}{v \cdot \frac{-1}{t1 + u}} \]

      2. frac-2neg 81.1%

         \[\leadsto v \cdot \color{blue}{\frac{--1}{-\left(t1 + u\right)}} \]

      3. metadata-eval 81.1%

         \[\leadsto v \cdot \frac{\color{blue}{1}}{-\left(t1 + u\right)} \]

      4. un-div-inv 81.3%

         \[\leadsto \color{blue}{\frac{v}{-\left(t1 + u\right)}} \]

      5. +-commutative 81.3%

         \[\leadsto \frac{v}{-\color{blue}{\left(u + t1\right)}} \]

      6. distribute-neg-in 81.3%

         \[\leadsto \frac{v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      7. add-sqr-sqrt 39.1%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)} \]

      8. sqrt-unprod 84.0%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \left(-t1\right)} \]

      9. sqr-neg 84.0%

         \[\leadsto \frac{v}{\sqrt{\color{blue}{u \cdot u}} + \left(-t1\right)} \]

      10. sqrt-unprod 42.9%

          \[\leadsto \frac{v}{\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \left(-t1\right)} \]

      11. add-sqr-sqrt 81.8%

          \[\leadsto \frac{v}{\color{blue}{u} + \left(-t1\right)} \]

   6. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{v}{u + \left(-t1\right)}} \]
   7. Step-by-step derivation

      1. unsub-neg 81.8%

         \[\leadsto \frac{v}{\color{blue}{u - t1}} \]

   8. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{v}{u - t1}} \]

   if -1.20000000000000009e-108 < t1 < 1.8e-72

   1. Initial program 80.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 79.6%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 79.6%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 79.6%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   4. Taylor expanded in t1 around 0 74.5%

      \[\leadsto v \cdot \color{blue}{\left(-1 \cdot \frac{t1}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 74.5%

         \[\leadsto v \cdot \color{blue}{\frac{-1 \cdot t1}{{u}^{2}}} \]

      2. neg-mul-1 74.5%

         \[\leadsto v \cdot \frac{\color{blue}{-t1}}{{u}^{2}} \]

      3. unpow2 74.5%

         \[\leadsto v \cdot \frac{-t1}{\color{blue}{u \cdot u}} \]

   6. Simplified 74.5%

      \[\leadsto v \cdot \color{blue}{\frac{-t1}{u \cdot u}} \]

   if 1.8e-72 < t1

   1. Initial program 57.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 73.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      2. associate-/l* 99.4%

         \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 79.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   5. Step-by-step derivation

      1. neg-mul-1 79.2%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   6. Simplified 79.2%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.8 \cdot 10^{-72}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 7: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{v \cdot \frac{-t1}{t1 + u}}{t1 + u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* v (/ (- t1) (+ t1 u))) (+ t1 u)))

C

double code(double u, double v, double t1) {
	return (v * (-t1 / (t1 + u))) / (t1 + u);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v * (-t1 / (t1 + u))) / (t1 + u)
end function

Java

public static double code(double u, double v, double t1) {
	return (v * (-t1 / (t1 + u))) / (t1 + u);
}

Python

def code(u, v, t1):
	return (v * (-t1 / (t1 + u))) / (t1 + u)

Julia

function code(u, v, t1)
	return Float64(Float64(v * Float64(Float64(-t1) / Float64(t1 + u))) / Float64(t1 + u))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v * (-t1 / (t1 + u))) / (t1 + u);
end

Wolfram

code[u_, v_, t1_] := N[(N[(v * N[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/r* 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

   2. associate-/l* 99.3%

      \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]

4. Taylor expanded in v around 0 81.4%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{t1 \cdot v}{t1 + u}}}{t1 + u} \]
5. Step-by-step derivation

   1. mul-1-neg 81.4%

      \[\leadsto \frac{\color{blue}{-\frac{t1 \cdot v}{t1 + u}}}{t1 + u} \]

   2. associate-*l/ 99.1%

      \[\leadsto \frac{-\color{blue}{\frac{t1}{t1 + u} \cdot v}}{t1 + u} \]

   3. distribute-rgt-neg-out 99.1%

      \[\leadsto \frac{\color{blue}{\frac{t1}{t1 + u} \cdot \left(-v\right)}}{t1 + u} \]

6. Simplified 99.1%

   \[\leadsto \frac{\color{blue}{\frac{t1}{t1 + u} \cdot \left(-v\right)}}{t1 + u} \]

7. Final simplification 99.1%

   \[\leadsto \frac{v \cdot \frac{-t1}{t1 + u}}{t1 + u} \]

Alternative 8: 68.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+124} \lor \neg \left(u \leq 2.5 \cdot 10^{+141}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.2e+124) (not (<= u 2.5e+141)))
   (* t1 (/ (/ v u) u))
   (/ v (- u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+124) || !(u <= 2.5e+141)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.2d+124)) .or. (.not. (u <= 2.5d+141))) then
        tmp = t1 * ((v / u) / u)
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+124) || !(u <= 2.5e+141)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.2e+124) or not (u <= 2.5e+141):
		tmp = t1 * ((v / u) / u)
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.2e+124) || !(u <= 2.5e+141))
		tmp = Float64(t1 * Float64(Float64(v / u) / u));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.2e+124) || ~((u <= 2.5e+141)))
		tmp = t1 * ((v / u) / u);
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.2e+124], N[Not[LessEqual[u, 2.5e+141]], $MachinePrecision]], N[(t1 * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.20000000000000003e124 or 2.50000000000000013e141 < u

   1. Initial program 70.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 72.3%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 72.3%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 72.3%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   4. Taylor expanded in t1 around 0 72.0%

      \[\leadsto v \cdot \color{blue}{\left(-1 \cdot \frac{t1}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 72.0%

         \[\leadsto v \cdot \color{blue}{\frac{-1 \cdot t1}{{u}^{2}}} \]

      2. neg-mul-1 72.0%

         \[\leadsto v \cdot \frac{\color{blue}{-t1}}{{u}^{2}} \]

      3. unpow2 72.0%

         \[\leadsto v \cdot \frac{-t1}{\color{blue}{u \cdot u}} \]

   6. Simplified 72.0%

      \[\leadsto v \cdot \color{blue}{\frac{-t1}{u \cdot u}} \]
   7. Step-by-step derivation

      1. add-log-exp 68.6%

         \[\leadsto \color{blue}{\log \left(e^{v \cdot \frac{-t1}{u \cdot u}}\right)} \]

      2. exp-prod 68.6%

         \[\leadsto \log \color{blue}{\left({\left(e^{v}\right)}^{\left(\frac{-t1}{u \cdot u}\right)}\right)} \]

      3. add-sqr-sqrt 58.8%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{u \cdot u}\right)}\right) \]

      4. sqrt-unprod 67.8%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{u \cdot u}\right)}\right) \]

      5. sqr-neg 67.8%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{u \cdot u}\right)}\right) \]

      6. sqrt-unprod 55.3%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{u \cdot u}\right)}\right) \]

      7. add-sqr-sqrt 68.5%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{t1}}{u \cdot u}\right)}\right) \]

   8. Applied egg-rr 68.5%

      \[\leadsto \color{blue}{\log \left({\left(e^{v}\right)}^{\left(\frac{t1}{u \cdot u}\right)}\right)} \]
   9. Step-by-step derivation

      1. log-pow 48.2%

         \[\leadsto \color{blue}{\frac{t1}{u \cdot u} \cdot \log \left(e^{v}\right)} \]

      2. rem-log-exp 68.5%

         \[\leadsto \frac{t1}{u \cdot u} \cdot \color{blue}{v} \]

   10. Simplified 68.5%

       \[\leadsto \color{blue}{\frac{t1}{u \cdot u} \cdot v} \]

   11. Taylor expanded in t1 around 0 68.1%

       \[\leadsto \color{blue}{\frac{t1 \cdot v}{{u}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 68.1%

          \[\leadsto \frac{t1 \cdot v}{\color{blue}{u \cdot u}} \]

       2. associate-*r/ 68.5%

          \[\leadsto \color{blue}{t1 \cdot \frac{v}{u \cdot u}} \]

       3. associate-/r* 68.2%

          \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{u}}{u}} \]

   13. Simplified 68.2%

       \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{u}}{u}} \]

   if -1.20000000000000003e124 < u < 2.50000000000000013e141

   1. Initial program 69.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 68.3%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      2. neg-mul-1 68.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. associate-*r/ 78.7%

         \[\leadsto \frac{-1 \cdot t1}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{v}}} \]

      4. times-frac 98.8%

         \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \frac{t1}{\frac{t1 + u}{v}}} \]

      5. div-inv 98.7%

         \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{\left(t1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

      6. clear-num 99.0%

         \[\leadsto \frac{-1}{t1 + u} \cdot \left(t1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \]

   3. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)} \]

   4. Taylor expanded in t1 around inf 70.2%

      \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{v} \]
   5. Step-by-step derivation

      1. *-commutative 70.2%

         \[\leadsto \color{blue}{v \cdot \frac{-1}{t1 + u}} \]

      2. frac-2neg 70.2%

         \[\leadsto v \cdot \color{blue}{\frac{--1}{-\left(t1 + u\right)}} \]

      3. metadata-eval 70.2%

         \[\leadsto v \cdot \frac{\color{blue}{1}}{-\left(t1 + u\right)} \]

      4. un-div-inv 70.5%

         \[\leadsto \color{blue}{\frac{v}{-\left(t1 + u\right)}} \]

      5. +-commutative 70.5%

         \[\leadsto \frac{v}{-\color{blue}{\left(u + t1\right)}} \]

      6. distribute-neg-in 70.5%

         \[\leadsto \frac{v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      7. add-sqr-sqrt 35.7%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)} \]

      8. sqrt-unprod 71.9%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \left(-t1\right)} \]

      9. sqr-neg 71.9%

         \[\leadsto \frac{v}{\sqrt{\color{blue}{u \cdot u}} + \left(-t1\right)} \]

      10. sqrt-unprod 36.1%

          \[\leadsto \frac{v}{\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \left(-t1\right)} \]

      11. add-sqr-sqrt 71.8%

          \[\leadsto \frac{v}{\color{blue}{u} + \left(-t1\right)} \]

   6. Applied egg-rr 71.8%

      \[\leadsto \color{blue}{\frac{v}{u + \left(-t1\right)}} \]
   7. Step-by-step derivation

      1. unsub-neg 71.8%

         \[\leadsto \frac{v}{\color{blue}{u - t1}} \]

   8. Simplified 71.8%

      \[\leadsto \color{blue}{\frac{v}{u - t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+124} \lor \neg \left(u \leq 2.5 \cdot 10^{+141}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 9: 68.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{+130} \lor \neg \left(u \leq 8.8 \cdot 10^{+141}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.9e+130) (not (<= u 8.8e+141)))
   (* v (/ t1 (* u u)))
   (/ v (- u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.9e+130) || !(u <= 8.8e+141)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.9d+130)) .or. (.not. (u <= 8.8d+141))) then
        tmp = v * (t1 / (u * u))
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.9e+130) || !(u <= 8.8e+141)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.9e+130) or not (u <= 8.8e+141):
		tmp = v * (t1 / (u * u))
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.9e+130) || !(u <= 8.8e+141))
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.9e+130) || ~((u <= 8.8e+141)))
		tmp = v * (t1 / (u * u));
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.9e+130], N[Not[LessEqual[u, 8.8e+141]], $MachinePrecision]], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.8999999999999999e130 or 8.8e141 < u

   1. Initial program 70.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 72.3%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 72.3%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 72.3%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   4. Taylor expanded in t1 around 0 72.0%

      \[\leadsto v \cdot \color{blue}{\left(-1 \cdot \frac{t1}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 72.0%

         \[\leadsto v \cdot \color{blue}{\frac{-1 \cdot t1}{{u}^{2}}} \]

      2. neg-mul-1 72.0%

         \[\leadsto v \cdot \frac{\color{blue}{-t1}}{{u}^{2}} \]

      3. unpow2 72.0%

         \[\leadsto v \cdot \frac{-t1}{\color{blue}{u \cdot u}} \]

   6. Simplified 72.0%

      \[\leadsto v \cdot \color{blue}{\frac{-t1}{u \cdot u}} \]
   7. Step-by-step derivation

      1. add-log-exp 68.6%

         \[\leadsto \color{blue}{\log \left(e^{v \cdot \frac{-t1}{u \cdot u}}\right)} \]

      2. exp-prod 68.6%

         \[\leadsto \log \color{blue}{\left({\left(e^{v}\right)}^{\left(\frac{-t1}{u \cdot u}\right)}\right)} \]

      3. add-sqr-sqrt 58.8%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{u \cdot u}\right)}\right) \]

      4. sqrt-unprod 67.8%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{u \cdot u}\right)}\right) \]

      5. sqr-neg 67.8%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{u \cdot u}\right)}\right) \]

      6. sqrt-unprod 55.3%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{u \cdot u}\right)}\right) \]

      7. add-sqr-sqrt 68.5%

         \[\leadsto \log \left({\left(e^{v}\right)}^{\left(\frac{\color{blue}{t1}}{u \cdot u}\right)}\right) \]

   8. Applied egg-rr 68.5%

      \[\leadsto \color{blue}{\log \left({\left(e^{v}\right)}^{\left(\frac{t1}{u \cdot u}\right)}\right)} \]
   9. Step-by-step derivation

      1. log-pow 48.2%

         \[\leadsto \color{blue}{\frac{t1}{u \cdot u} \cdot \log \left(e^{v}\right)} \]

      2. rem-log-exp 68.5%

         \[\leadsto \frac{t1}{u \cdot u} \cdot \color{blue}{v} \]

   10. Simplified 68.5%

       \[\leadsto \color{blue}{\frac{t1}{u \cdot u} \cdot v} \]

   if -2.8999999999999999e130 < u < 8.8e141

   1. Initial program 69.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 68.3%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      2. neg-mul-1 68.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. associate-*r/ 78.7%

         \[\leadsto \frac{-1 \cdot t1}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{v}}} \]

      4. times-frac 98.8%

         \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \frac{t1}{\frac{t1 + u}{v}}} \]

      5. div-inv 98.7%

         \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{\left(t1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

      6. clear-num 99.0%

         \[\leadsto \frac{-1}{t1 + u} \cdot \left(t1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \]

   3. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)} \]

   4. Taylor expanded in t1 around inf 70.2%

      \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{v} \]
   5. Step-by-step derivation

      1. *-commutative 70.2%

         \[\leadsto \color{blue}{v \cdot \frac{-1}{t1 + u}} \]

      2. frac-2neg 70.2%

         \[\leadsto v \cdot \color{blue}{\frac{--1}{-\left(t1 + u\right)}} \]

      3. metadata-eval 70.2%

         \[\leadsto v \cdot \frac{\color{blue}{1}}{-\left(t1 + u\right)} \]

      4. un-div-inv 70.5%

         \[\leadsto \color{blue}{\frac{v}{-\left(t1 + u\right)}} \]

      5. +-commutative 70.5%

         \[\leadsto \frac{v}{-\color{blue}{\left(u + t1\right)}} \]

      6. distribute-neg-in 70.5%

         \[\leadsto \frac{v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      7. add-sqr-sqrt 35.7%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)} \]

      8. sqrt-unprod 71.9%

         \[\leadsto \frac{v}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \left(-t1\right)} \]

      9. sqr-neg 71.9%

         \[\leadsto \frac{v}{\sqrt{\color{blue}{u \cdot u}} + \left(-t1\right)} \]

      10. sqrt-unprod 36.1%

          \[\leadsto \frac{v}{\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \left(-t1\right)} \]

      11. add-sqr-sqrt 71.8%

          \[\leadsto \frac{v}{\color{blue}{u} + \left(-t1\right)} \]

   6. Applied egg-rr 71.8%

      \[\leadsto \color{blue}{\frac{v}{u + \left(-t1\right)}} \]
   7. Step-by-step derivation

      1. unsub-neg 71.8%

         \[\leadsto \frac{v}{\color{blue}{u - t1}} \]

   8. Simplified 71.8%

      \[\leadsto \color{blue}{\frac{v}{u - t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{+130} \lor \neg \left(u \leq 8.8 \cdot 10^{+141}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 10: 57.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+202} \lor \neg \left(u \leq 1.8 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2e+202) (not (<= u 1.8e+143))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2e+202) || !(u <= 1.8e+143)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2d+202)) .or. (.not. (u <= 1.8d+143))) then
        tmp = -v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2e+202) || !(u <= 1.8e+143)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2e+202) or not (u <= 1.8e+143):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2e+202) || !(u <= 1.8e+143))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2e+202) || ~((u <= 1.8e+143)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2e+202], N[Not[LessEqual[u, 1.8e+143]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.9999999999999998e202 or 1.8e143 < u

   1. Initial program 78.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 91.4%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      2. associate-/l* 99.8%

         \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}} \]

   4. Taylor expanded in t1 around 0 89.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{t1 \cdot v}{u}}}{t1 + u} \]
   5. Step-by-step derivation

      1. mul-1-neg 89.7%

         \[\leadsto \frac{\color{blue}{-\frac{t1 \cdot v}{u}}}{t1 + u} \]

      2. associate-/l* 95.2%

         \[\leadsto \frac{-\color{blue}{\frac{t1}{\frac{u}{v}}}}{t1 + u} \]

      3. distribute-neg-frac 95.2%

         \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{u}{v}}}}{t1 + u} \]

   6. Simplified 95.2%

      \[\leadsto \frac{\color{blue}{\frac{-t1}{\frac{u}{v}}}}{t1 + u} \]

   7. Taylor expanded in t1 around inf 50.6%

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

      1. neg-mul-1 50.6%

         \[\leadsto \color{blue}{-\frac{v}{u}} \]

      2. distribute-neg-frac 50.6%

         \[\leadsto \color{blue}{\frac{-v}{u}} \]

   9. Simplified 50.6%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]

   if -1.9999999999999998e202 < u < 1.8e143

   1. Initial program 67.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 71.3%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 71.3%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   4. Taylor expanded in t1 around inf 66.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   5. Step-by-step derivation

      1. associate-*r/ 66.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 66.3%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   6. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+202} \lor \neg \left(u \leq 1.8 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 60.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- u t1)))

C

double code(double u, double v, double t1) {
	return v / (u - t1);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / (u - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / (u - t1);
}

Python

def code(u, v, t1):
	return v / (u - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(u - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / (u - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/l* 69.6%

      \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

   2. neg-mul-1 69.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

   3. associate-*r/ 81.5%

      \[\leadsto \frac{-1 \cdot t1}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{v}}} \]

   4. times-frac 99.1%

      \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \frac{t1}{\frac{t1 + u}{v}}} \]

   5. div-inv 99.0%

      \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{\left(t1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

   6. clear-num 99.2%

      \[\leadsto \frac{-1}{t1 + u} \cdot \left(t1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \]

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{-1}{t1 + u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)} \]

4. Taylor expanded in t1 around inf 63.3%

   \[\leadsto \frac{-1}{t1 + u} \cdot \color{blue}{v} \]
5. Step-by-step derivation

   1. *-commutative 63.3%

      \[\leadsto \color{blue}{v \cdot \frac{-1}{t1 + u}} \]

   2. frac-2neg 63.3%

      \[\leadsto v \cdot \color{blue}{\frac{--1}{-\left(t1 + u\right)}} \]

   3. metadata-eval 63.3%

      \[\leadsto v \cdot \frac{\color{blue}{1}}{-\left(t1 + u\right)} \]

   4. un-div-inv 63.5%

      \[\leadsto \color{blue}{\frac{v}{-\left(t1 + u\right)}} \]

   5. +-commutative 63.5%

      \[\leadsto \frac{v}{-\color{blue}{\left(u + t1\right)}} \]

   6. distribute-neg-in 63.5%

      \[\leadsto \frac{v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   7. add-sqr-sqrt 32.4%

      \[\leadsto \frac{v}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)} \]

   8. sqrt-unprod 70.1%

      \[\leadsto \frac{v}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \left(-t1\right)} \]

   9. sqr-neg 70.1%

      \[\leadsto \frac{v}{\sqrt{\color{blue}{u \cdot u}} + \left(-t1\right)} \]

   10. sqrt-unprod 31.8%

       \[\leadsto \frac{v}{\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \left(-t1\right)} \]

   11. add-sqr-sqrt 64.3%

       \[\leadsto \frac{v}{\color{blue}{u} + \left(-t1\right)} \]

6. Applied egg-rr 64.3%

   \[\leadsto \color{blue}{\frac{v}{u + \left(-t1\right)}} \]
7. Step-by-step derivation

   1. unsub-neg 64.3%

      \[\leadsto \frac{v}{\color{blue}{u - t1}} \]

8. Simplified 64.3%

   \[\leadsto \color{blue}{\frac{v}{u - t1}} \]

9. Final simplification 64.3%

   \[\leadsto \frac{v}{u - t1} \]

Alternative 12: 53.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{-v}{t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (- v) t1))

C

double code(double u, double v, double t1) {
	return -v / t1;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = -v / t1
end function

Java

public static double code(double u, double v, double t1) {
	return -v / t1;
}

Python

def code(u, v, t1):
	return -v / t1

Julia

function code(u, v, t1)
	return Float64(Float64(-v) / t1)
end

MATLAB

function tmp = code(u, v, t1)
	tmp = -v / t1;
end

Wolfram

code[u_, v_, t1_] := N[((-v) / t1), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-*l/ 73.3%

      \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

   2. *-commutative 73.3%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

3. Simplified 73.3%

   \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

4. Taylor expanded in t1 around inf 56.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
5. Step-by-step derivation

   1. associate-*r/ 56.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

   2. neg-mul-1 56.7%

      \[\leadsto \frac{\color{blue}{-v}}{t1} \]

6. Simplified 56.7%

   \[\leadsto \color{blue}{\frac{-v}{t1}} \]

7. Final simplification 56.7%

   \[\leadsto \frac{-v}{t1} \]

Alternative 13: 13.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v t1))

C

double code(double u, double v, double t1) {
	return v / t1;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / t1
end function

Java

public static double code(double u, double v, double t1) {
	return v / t1;
}

Python

def code(u, v, t1):
	return v / t1

Julia

function code(u, v, t1)
	return Float64(v / t1)
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / t1;
end

Wolfram

code[u_, v_, t1_] := N[(v / t1), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-*l/ 73.3%

      \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

   2. *-commutative 73.3%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

3. Simplified 73.3%

   \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

4. Taylor expanded in t1 around inf 56.6%

   \[\leadsto v \cdot \color{blue}{\frac{-1}{t1}} \]
5. Step-by-step derivation

   1. expm1-log1p-u 46.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \frac{-1}{t1}\right)\right)} \]

   2. expm1-udef 30.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(v \cdot \frac{-1}{t1}\right)} - 1} \]

   3. *-commutative 30.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{t1} \cdot v}\right)} - 1 \]

   4. associate-*l/ 30.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1 \cdot v}{t1}}\right)} - 1 \]

   5. neg-mul-1 30.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-v}}{t1}\right)} - 1 \]

   6. add-sqr-sqrt 15.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1}\right)} - 1 \]

   7. sqrt-unprod 27.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{t1}\right)} - 1 \]

   8. sqr-neg 27.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{v \cdot v}}}{t1}\right)} - 1 \]

   9. sqrt-unprod 11.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1}\right)} - 1 \]

   10. add-sqr-sqrt 22.6%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{v}}{t1}\right)} - 1 \]

6. Applied egg-rr 22.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v}{t1}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 15.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{t1}\right)\right)} \]

   2. expm1-log1p 15.8%

      \[\leadsto \color{blue}{\frac{v}{t1}} \]

8. Simplified 15.8%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]

9. Final simplification 15.8%

   \[\leadsto \frac{v}{t1} \]

Reproduce

herbie shell --seed 2023278 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))