Rosa's DopplerBench

Percentage Accurate: 72.5% → 97.9%

Time: 13.9s

Alternatives: 12

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: 72.5% 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

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

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(t1 + u)) * Float64(-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[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * (-v)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

   1. associate-/r* 81.5%

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

   2. associate-/l* 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in v around 0 81.5%

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

   1. mul-1-neg 81.5%

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

   2. associate-*l/ 98.2%

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

   3. distribute-rgt-neg-out 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

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

Alternative 2: 90.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -1 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.7 \cdot 10^{-264}:\\ \;\;\;\;\frac{v}{u \cdot \frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq 5.5 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{u}{\frac{t1}{v}} - v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -1.35e+154)
     (/ (- v) t1)
     (if (<= t1 -1e-203)
       t_1
       (if (<= t1 1.7e-264)
         (/ v (* u (/ (- u) t1)))
         (if (<= t1 5.5e+155) t_1 (/ (- (/ u (/ t1 v)) v) (+ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.35e+154) {
		tmp = -v / t1;
	} else if (t1 <= -1e-203) {
		tmp = t_1;
	} else if (t1 <= 1.7e-264) {
		tmp = v / (u * (-u / t1));
	} else if (t1 <= 5.5e+155) {
		tmp = t_1;
	} else {
		tmp = ((u / (t1 / v)) - 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) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-1.35d+154)) then
        tmp = -v / t1
    else if (t1 <= (-1d-203)) then
        tmp = t_1
    else if (t1 <= 1.7d-264) then
        tmp = v / (u * (-u / t1))
    else if (t1 <= 5.5d+155) then
        tmp = t_1
    else
        tmp = ((u / (t1 / v)) - v) / (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 <= -1.35e+154) {
		tmp = -v / t1;
	} else if (t1 <= -1e-203) {
		tmp = t_1;
	} else if (t1 <= 1.7e-264) {
		tmp = v / (u * (-u / t1));
	} else if (t1 <= 5.5e+155) {
		tmp = t_1;
	} else {
		tmp = ((u / (t1 / v)) - v) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -1.35e+154:
		tmp = -v / t1
	elif t1 <= -1e-203:
		tmp = t_1
	elif t1 <= 1.7e-264:
		tmp = v / (u * (-u / t1))
	elif t1 <= 5.5e+155:
		tmp = t_1
	else:
		tmp = ((u / (t1 / v)) - v) / (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 <= -1.35e+154)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= -1e-203)
		tmp = t_1;
	elseif (t1 <= 1.7e-264)
		tmp = Float64(v / Float64(u * Float64(Float64(-u) / t1)));
	elseif (t1 <= 5.5e+155)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(u / Float64(t1 / v)) - v) / 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 <= -1.35e+154)
		tmp = -v / t1;
	elseif (t1 <= -1e-203)
		tmp = t_1;
	elseif (t1 <= 1.7e-264)
		tmp = v / (u * (-u / t1));
	elseif (t1 <= 5.5e+155)
		tmp = t_1;
	else
		tmp = ((u / (t1 / v)) - v) / (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, -1.35e+154], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, -1e-203], t$95$1, If[LessEqual[t1, 1.7e-264], N[(v / N[(u * N[((-u) / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.5e+155], t$95$1, N[(N[(N[(u / N[(t1 / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.35000000000000003e154

   1. Initial program 34.5%

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

      1. associate-*l/ 36.1%

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

      2. *-commutative 36.1%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in t1 around inf 87.3%

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

      1. associate-*r/ 87.3%

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

      2. neg-mul-1 87.3%

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

   6. Simplified 87.3%

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

   if -1.35000000000000003e154 < t1 < -1e-203 or 1.6999999999999999e-264 < t1 < 5.5000000000000001e155

   1. Initial program 84.2%

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

      1. associate-*l/ 92.1%

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

      2. *-commutative 92.1%

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

   3. Simplified 92.1%

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

   if -1e-203 < t1 < 1.6999999999999999e-264

   1. Initial program 72.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 79.6%

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

      1. associate-*r/ 79.6%

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

      2. neg-mul-1 79.6%

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

      3. unpow2 79.6%

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

   6. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. un-div-inv 79.6%

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

      3. associate-/l* 93.8%

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

      4. add-sqr-sqrt 57.6%

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

      5. sqrt-unprod 51.5%

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

      6. sqr-neg 51.5%

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

      7. sqrt-unprod 16.1%

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

      8. add-sqr-sqrt 50.8%

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

   8. Applied egg-rr 50.8%

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

      1. frac-2neg 50.8%

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

      2. distribute-frac-neg 50.8%

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

      3. distribute-frac-neg 50.8%

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

      4. add-sqr-sqrt 34.6%

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

      5. sqrt-unprod 51.5%

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

      6. sqr-neg 51.5%

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

      7. sqrt-unprod 36.0%

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

      8. add-sqr-sqrt 93.8%

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

      9. associate-/r/ 93.9%

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

      10. distribute-rgt-neg-in 93.9%

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

   10. Applied egg-rr 93.9%

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

   if 5.5000000000000001e155 < t1

   1. Initial program 51.5%

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

      1. associate-/r* 70.9%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 93.4%

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

      1. neg-mul-1 93.4%

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

      2. +-commutative 93.4%

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

      3. unsub-neg 93.4%

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

      4. associate-/l* 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -1 \cdot 10^{-203}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.7 \cdot 10^{-264}:\\ \;\;\;\;\frac{v}{u \cdot \frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq 5.5 \cdot 10^{+155}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{u}{\frac{t1}{v}} - v}{t1 + u}\\ \end{array} \]

Alternative 3: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1}\\ \mathbf{if}\;u \leq -5.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;u \leq 1.95 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 5.1 \cdot 10^{-23}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) t1)))
   (if (<= u -5.5e-40)
     (/ (/ (- t1) (/ u v)) (+ t1 u))
     (if (<= u 1.95e-51)
       t_1
       (if (<= u 5.1e-23)
         (/ (* v (/ (- t1) u)) (+ t1 u))
         (if (<= u 3.5e+68) t_1 (/ (* t1 (/ v u)) (- t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / t1;
	double tmp;
	if (u <= -5.5e-40) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (u <= 1.95e-51) {
		tmp = t_1;
	} else if (u <= 5.1e-23) {
		tmp = (v * (-t1 / u)) / (t1 + u);
	} else if (u <= 3.5e+68) {
		tmp = t_1;
	} else {
		tmp = (t1 * (v / u)) / (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
    if (u <= (-5.5d-40)) then
        tmp = (-t1 / (u / v)) / (t1 + u)
    else if (u <= 1.95d-51) then
        tmp = t_1
    else if (u <= 5.1d-23) then
        tmp = (v * (-t1 / u)) / (t1 + u)
    else if (u <= 3.5d+68) then
        tmp = t_1
    else
        tmp = (t1 * (v / u)) / (t1 - u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / t1;
	double tmp;
	if (u <= -5.5e-40) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (u <= 1.95e-51) {
		tmp = t_1;
	} else if (u <= 5.1e-23) {
		tmp = (v * (-t1 / u)) / (t1 + u);
	} else if (u <= 3.5e+68) {
		tmp = t_1;
	} else {
		tmp = (t1 * (v / u)) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / t1
	tmp = 0
	if u <= -5.5e-40:
		tmp = (-t1 / (u / v)) / (t1 + u)
	elif u <= 1.95e-51:
		tmp = t_1
	elif u <= 5.1e-23:
		tmp = (v * (-t1 / u)) / (t1 + u)
	elif u <= 3.5e+68:
		tmp = t_1
	else:
		tmp = (t1 * (v / u)) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / t1)
	tmp = 0.0
	if (u <= -5.5e-40)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / Float64(t1 + u));
	elseif (u <= 1.95e-51)
		tmp = t_1;
	elseif (u <= 5.1e-23)
		tmp = Float64(Float64(v * Float64(Float64(-t1) / u)) / Float64(t1 + u));
	elseif (u <= 3.5e+68)
		tmp = t_1;
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / t1;
	tmp = 0.0;
	if (u <= -5.5e-40)
		tmp = (-t1 / (u / v)) / (t1 + u);
	elseif (u <= 1.95e-51)
		tmp = t_1;
	elseif (u <= 5.1e-23)
		tmp = (v * (-t1 / u)) / (t1 + u);
	elseif (u <= 3.5e+68)
		tmp = t_1;
	else
		tmp = (t1 * (v / u)) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / t1), $MachinePrecision]}, If[LessEqual[u, -5.5e-40], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.95e-51], t$95$1, If[LessEqual[u, 5.1e-23], N[(N[(v * N[((-t1) / u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.5e+68], t$95$1, N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -5.50000000000000002e-40

   1. Initial program 83.3%

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

      1. associate-/r* 95.4%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 82.3%

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

      1. mul-1-neg 82.3%

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

      2. associate-/l* 83.8%

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

      3. distribute-neg-frac 83.8%

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

   6. Simplified 83.8%

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

   if -5.50000000000000002e-40 < u < 1.9499999999999999e-51 or 5.10000000000000011e-23 < u < 3.49999999999999977e68

   1. Initial program 62.2%

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

      1. associate-*l/ 75.4%

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

      2. *-commutative 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in t1 around inf 81.8%

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

      1. associate-*r/ 81.8%

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

      2. neg-mul-1 81.8%

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

   6. Simplified 81.8%

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

   if 1.9499999999999999e-51 < u < 5.10000000000000011e-23

   1. Initial program 89.5%

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

      1. associate-/r* 99.2%

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

      2. associate-/l* 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in v around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      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. Taylor expanded in t1 around 0 72.1%

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

   if 3.49999999999999977e68 < u

   1. Initial program 75.8%

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

      1. associate-/r* 87.0%

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

      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. Taylor expanded in t1 around 0 86.9%

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

      1. mul-1-neg 86.9%

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

      2. associate-/l* 92.4%

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

      3. distribute-neg-frac 92.4%

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

   6. Simplified 92.4%

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

      1. frac-2neg 92.4%

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

      2. distribute-frac-neg 92.4%

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

      3. associate-/l* 86.9%

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

      4. mul-1-neg 86.9%

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

      5. div-inv 87.0%

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

      6. mul-1-neg 87.0%

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

      7. remove-double-neg 87.0%

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

      8. div-inv 86.8%

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

      9. associate-*l* 92.3%

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

      10. div-inv 92.4%

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

      11. distribute-neg-in 92.4%

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

      12. add-sqr-sqrt 49.0%

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

      13. sqrt-unprod 87.4%

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

      14. sqr-neg 87.4%

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

      15. sqrt-unprod 43.3%

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

      16. add-sqr-sqrt 92.8%

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

      17. sub-neg 92.8%

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

   8. Applied egg-rr 92.8%

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

      1. associate-*r/ 92.8%

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

      2. *-rgt-identity 92.8%

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

   10. Simplified 92.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;u \leq 1.95 \cdot 10^{-51}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 5.1 \cdot 10^{-23}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \]

Alternative 4: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1}\\ \mathbf{if}\;u \leq -5.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;u \leq 3 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 4.2 \cdot 10^{-24}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{t1 + u}}{t1 - u}\\ \mathbf{elif}\;u \leq 4.2 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) t1)))
   (if (<= u -5.5e-40)
     (/ (/ (- t1) (/ u v)) (+ t1 u))
     (if (<= u 3e-51)
       t_1
       (if (<= u 4.2e-24)
         (* v (/ (/ t1 (+ t1 u)) (- t1 u)))
         (if (<= u 4.2e+63) t_1 (/ (* t1 (/ v u)) (- t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / t1;
	double tmp;
	if (u <= -5.5e-40) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (u <= 3e-51) {
		tmp = t_1;
	} else if (u <= 4.2e-24) {
		tmp = v * ((t1 / (t1 + u)) / (t1 - u));
	} else if (u <= 4.2e+63) {
		tmp = t_1;
	} else {
		tmp = (t1 * (v / u)) / (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
    if (u <= (-5.5d-40)) then
        tmp = (-t1 / (u / v)) / (t1 + u)
    else if (u <= 3d-51) then
        tmp = t_1
    else if (u <= 4.2d-24) then
        tmp = v * ((t1 / (t1 + u)) / (t1 - u))
    else if (u <= 4.2d+63) then
        tmp = t_1
    else
        tmp = (t1 * (v / u)) / (t1 - u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / t1;
	double tmp;
	if (u <= -5.5e-40) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (u <= 3e-51) {
		tmp = t_1;
	} else if (u <= 4.2e-24) {
		tmp = v * ((t1 / (t1 + u)) / (t1 - u));
	} else if (u <= 4.2e+63) {
		tmp = t_1;
	} else {
		tmp = (t1 * (v / u)) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / t1
	tmp = 0
	if u <= -5.5e-40:
		tmp = (-t1 / (u / v)) / (t1 + u)
	elif u <= 3e-51:
		tmp = t_1
	elif u <= 4.2e-24:
		tmp = v * ((t1 / (t1 + u)) / (t1 - u))
	elif u <= 4.2e+63:
		tmp = t_1
	else:
		tmp = (t1 * (v / u)) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / t1)
	tmp = 0.0
	if (u <= -5.5e-40)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / Float64(t1 + u));
	elseif (u <= 3e-51)
		tmp = t_1;
	elseif (u <= 4.2e-24)
		tmp = Float64(v * Float64(Float64(t1 / Float64(t1 + u)) / Float64(t1 - u)));
	elseif (u <= 4.2e+63)
		tmp = t_1;
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / t1;
	tmp = 0.0;
	if (u <= -5.5e-40)
		tmp = (-t1 / (u / v)) / (t1 + u);
	elseif (u <= 3e-51)
		tmp = t_1;
	elseif (u <= 4.2e-24)
		tmp = v * ((t1 / (t1 + u)) / (t1 - u));
	elseif (u <= 4.2e+63)
		tmp = t_1;
	else
		tmp = (t1 * (v / u)) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / t1), $MachinePrecision]}, If[LessEqual[u, -5.5e-40], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3e-51], t$95$1, If[LessEqual[u, 4.2e-24], N[(v * N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.2e+63], t$95$1, N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -5.50000000000000002e-40

   1. Initial program 83.3%

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

      1. associate-/r* 95.4%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 82.3%

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

      1. mul-1-neg 82.3%

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

      2. associate-/l* 83.8%

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

      3. distribute-neg-frac 83.8%

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

   6. Simplified 83.8%

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

   if -5.50000000000000002e-40 < u < 3.00000000000000002e-51 or 4.1999999999999999e-24 < u < 4.2000000000000004e63

   1. Initial program 62.2%

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

      1. associate-*l/ 75.4%

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

      2. *-commutative 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in t1 around inf 81.8%

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

      1. associate-*r/ 81.8%

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

      2. neg-mul-1 81.8%

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

   6. Simplified 81.8%

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

   if 3.00000000000000002e-51 < u < 4.1999999999999999e-24

   1. Initial program 89.5%

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

      1. associate-*l/ 89.8%

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

      2. *-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. associate-/r* 99.1%

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

      2. associate-*r/ 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-/r/ 89.8%

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

      5. div-inv 89.8%

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

      6. frac-2neg 89.8%

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

      7. frac-times 89.8%

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

      8. remove-double-neg 89.8%

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

      9. *-commutative 89.8%

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

      10. *-un-lft-identity 89.8%

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

      11. distribute-neg-frac 89.8%

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

      12. distribute-neg-in 89.8%

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

      13. add-sqr-sqrt 49.7%

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

      14. sqrt-unprod 72.4%

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

      15. sqr-neg 72.4%

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

      16. sqrt-unprod 32.3%

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

      17. add-sqr-sqrt 72.3%

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

      18. sub-neg 72.3%

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

   5. Applied egg-rr 72.3%

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

      1. associate-/l/ 72.6%

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

      2. associate-/r/ 81.7%

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

   7. Simplified 81.7%

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

   if 4.2000000000000004e63 < u

   1. Initial program 75.8%

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

      1. associate-/r* 87.0%

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

      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. Taylor expanded in t1 around 0 86.9%

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

      1. mul-1-neg 86.9%

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

      2. associate-/l* 92.4%

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

      3. distribute-neg-frac 92.4%

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

   6. Simplified 92.4%

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

      1. frac-2neg 92.4%

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

      2. distribute-frac-neg 92.4%

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

      3. associate-/l* 86.9%

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

      4. mul-1-neg 86.9%

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

      5. div-inv 87.0%

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

      6. mul-1-neg 87.0%

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

      7. remove-double-neg 87.0%

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

      8. div-inv 86.8%

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

      9. associate-*l* 92.3%

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

      10. div-inv 92.4%

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

      11. distribute-neg-in 92.4%

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

      12. add-sqr-sqrt 49.0%

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

      13. sqrt-unprod 87.4%

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

      14. sqr-neg 87.4%

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

      15. sqrt-unprod 43.3%

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

      16. add-sqr-sqrt 92.8%

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

      17. sub-neg 92.8%

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

   8. Applied egg-rr 92.8%

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

      1. associate-*r/ 92.8%

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

      2. *-rgt-identity 92.8%

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

   10. Simplified 92.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;u \leq 3 \cdot 10^{-51}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4.2 \cdot 10^{-24}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{t1 + u}}{t1 - u}\\ \mathbf{elif}\;u \leq 4.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \]

Alternative 5: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{+47} \lor \neg \left(t1 \leq 3.55 \cdot 10^{-168} \lor \neg \left(t1 \leq 3.7 \cdot 10^{-47}\right) \land t1 \leq 1.4 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5e+47)
         (not
          (or (<= t1 3.55e-168) (and (not (<= t1 3.7e-47)) (<= t1 1.4e+24)))))
   (/ (- v) (+ t1 u))
   (* (/ v u) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5e+47) || !((t1 <= 3.55e-168) || (!(t1 <= 3.7e-47) && (t1 <= 1.4e+24)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (v / u) * (-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 <= (-5d+47)) .or. (.not. (t1 <= 3.55d-168) .or. (.not. (t1 <= 3.7d-47)) .and. (t1 <= 1.4d+24))) then
        tmp = -v / (t1 + u)
    else
        tmp = (v / u) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5e+47) || !((t1 <= 3.55e-168) || (!(t1 <= 3.7e-47) && (t1 <= 1.4e+24)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5e+47) or not ((t1 <= 3.55e-168) or (not (t1 <= 3.7e-47) and (t1 <= 1.4e+24))):
		tmp = -v / (t1 + u)
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5e+47) || !((t1 <= 3.55e-168) || (!(t1 <= 3.7e-47) && (t1 <= 1.4e+24))))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5e+47) || ~(((t1 <= 3.55e-168) || (~((t1 <= 3.7e-47)) && (t1 <= 1.4e+24)))))
		tmp = -v / (t1 + u);
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5e+47], N[Not[Or[LessEqual[t1, 3.55e-168], And[N[Not[LessEqual[t1, 3.7e-47]], $MachinePrecision], LessEqual[t1, 1.4e+24]]]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.00000000000000022e47 or 3.55000000000000009e-168 < t1 < 3.7e-47 or 1.4000000000000001e24 < t1

   1. Initial program 61.5%

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

      1. associate-/r* 74.0%

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

      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 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 -5.00000000000000022e47 < t1 < 3.55000000000000009e-168 or 3.7e-47 < t1 < 1.4000000000000001e24

   1. Initial program 84.8%

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

      1. associate-/r* 91.2%

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

      2. associate-/l* 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in v around 0 91.2%

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

      1. mul-1-neg 91.2%

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

      2. associate-*l/ 96.1%

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

      3. distribute-rgt-neg-out 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in t1 around 0 74.2%

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

      1. mul-1-neg 74.2%

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

      2. unpow2 74.2%

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

      3. times-frac 83.2%

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

   9. Simplified 83.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{+47} \lor \neg \left(t1 \leq 3.55 \cdot 10^{-168} \lor \neg \left(t1 \leq 3.7 \cdot 10^{-47}\right) \land t1 \leq 1.4 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 6: 95.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 1.7e+133)
   (* (- v) (/ (/ t1 (+ t1 u)) (+ t1 u)))
   (/ (* t1 (/ v u)) (- t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.7e+133) {
		tmp = -v * ((t1 / (t1 + u)) / (t1 + u));
	} else {
		tmp = (t1 * (v / u)) / (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 (u <= 1.7d+133) then
        tmp = -v * ((t1 / (t1 + u)) / (t1 + u))
    else
        tmp = (t1 * (v / u)) / (t1 - u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.7e+133) {
		tmp = -v * ((t1 / (t1 + u)) / (t1 + u));
	} else {
		tmp = (t1 * (v / u)) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 1.7e+133:
		tmp = -v * ((t1 / (t1 + u)) / (t1 + u))
	else:
		tmp = (t1 * (v / u)) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 1.7e+133)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / Float64(t1 + u)) / Float64(t1 + u)));
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 1.7e+133)
		tmp = -v * ((t1 / (t1 + u)) / (t1 + u));
	else
		tmp = (t1 * (v / u)) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 1.7e+133], N[((-v) * N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 1.69999999999999994e133

   1. Initial program 71.0%

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

      1. associate-*l/ 78.8%

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

      2. *-commutative 78.8%

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

   3. Simplified 78.8%

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

      1. neg-mul-1 78.8%

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

      2. times-frac 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. associate-*l/ 98.1%

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

      2. mul-1-neg 98.1%

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

   7. Simplified 98.1%

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

   if 1.69999999999999994e133 < u

   1. Initial program 74.8%

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

      1. associate-/r* 90.0%

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

      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 89.9%

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

      1. mul-1-neg 89.9%

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

      2. associate-/l* 94.9%

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

      3. distribute-neg-frac 94.9%

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

   6. Simplified 94.9%

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

      1. frac-2neg 94.9%

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

      2. distribute-frac-neg 94.9%

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

      3. associate-/l* 89.9%

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

      4. mul-1-neg 89.9%

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

      5. div-inv 89.9%

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

      6. mul-1-neg 89.9%

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

      7. remove-double-neg 89.9%

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

      8. div-inv 89.8%

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

      9. associate-*l* 94.8%

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

      10. div-inv 94.8%

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

      11. distribute-neg-in 94.8%

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

      12. add-sqr-sqrt 53.8%

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

      13. sqrt-unprod 88.0%

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

      14. sqr-neg 88.0%

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

      15. sqrt-unprod 41.0%

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

      16. add-sqr-sqrt 95.2%

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

      17. sub-neg 95.2%

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

   8. Applied egg-rr 95.2%

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

      1. associate-*r/ 95.3%

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

      2. *-rgt-identity 95.3%

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

   10. Simplified 95.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \]

Alternative 7: 68.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.25 \cdot 10^{+105} \lor \neg \left(u \leq 8.5 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.25e+105) (not (<= u 8.5e+62)))
   (/ t1 (* u (/ u v)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.25e+105) || !(u <= 8.5e+62)) {
		tmp = t1 / (u * (u / v));
	} 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 <= (-2.25d+105)) .or. (.not. (u <= 8.5d+62))) then
        tmp = t1 / (u * (u / v))
    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 <= -2.25e+105) || !(u <= 8.5e+62)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.25e+105) or not (u <= 8.5e+62):
		tmp = t1 / (u * (u / v))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.25e+105) || !(u <= 8.5e+62))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.25e+105) || ~((u <= 8.5e+62)))
		tmp = t1 / (u * (u / v));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.25e+105], N[Not[LessEqual[u, 8.5e+62]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.2500000000000001e105 or 8.4999999999999997e62 < u

   1. Initial program 80.5%

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

      1. associate-*l/ 80.5%

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

      2. *-commutative 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in t1 around 0 75.5%

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

      1. associate-*r/ 75.5%

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

      2. neg-mul-1 75.5%

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

      3. unpow2 75.5%

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

   6. Simplified 75.5%

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

      1. clear-num 74.9%

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

      2. un-div-inv 74.9%

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

      3. associate-/l* 78.1%

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

      4. add-sqr-sqrt 39.9%

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

      5. sqrt-unprod 59.9%

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

      6. sqr-neg 59.9%

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

      7. sqrt-unprod 33.2%

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

      8. add-sqr-sqrt 67.5%

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

   8. Applied egg-rr 67.5%

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

   9. Taylor expanded in v around 0 68.3%

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

       1. associate-/l* 73.0%

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

       2. unpow2 73.0%

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

       3. associate-*r/ 72.8%

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

   11. Simplified 72.8%

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

   if -2.2500000000000001e105 < u < 8.4999999999999997e62

   1. Initial program 66.1%

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

      1. associate-*l/ 76.1%

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

      2. *-commutative 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in t1 around inf 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

   6. Simplified 72.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.25 \cdot 10^{+105} \lor \neg \left(u \leq 8.5 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+183}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.1e+183) (/ v u) (if (<= u 1.8e+110) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+183) {
		tmp = v / u;
	} else if (u <= 1.8e+110) {
		tmp = -v / t1;
	} else {
		tmp = v / 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 (u <= (-2.1d+183)) then
        tmp = v / u
    else if (u <= 1.8d+110) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+183) {
		tmp = v / u;
	} else if (u <= 1.8e+110) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.1e+183:
		tmp = v / u
	elif u <= 1.8e+110:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.1e+183)
		tmp = Float64(v / u);
	elseif (u <= 1.8e+110)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.1e+183)
		tmp = v / u;
	elseif (u <= 1.8e+110)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.1e+183], N[(v / u), $MachinePrecision], If[LessEqual[u, 1.8e+110], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.1e183 or 1.7999999999999998e110 < u

   1. Initial program 79.9%

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

      1. associate-/r* 92.1%

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

      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 92.1%

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

      1. mul-1-neg 92.1%

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

      2. associate-/l* 97.3%

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

      3. distribute-neg-frac 97.3%

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

   6. Simplified 97.3%

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

      1. frac-2neg 97.3%

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

      2. distribute-frac-neg 97.3%

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

      3. associate-/l* 92.1%

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

      4. mul-1-neg 92.1%

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

      5. div-inv 92.1%

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

      6. mul-1-neg 92.1%

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

      7. remove-double-neg 92.1%

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

      8. div-inv 92.0%

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

      9. associate-*l* 97.2%

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

      10. div-inv 97.2%

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

      11. distribute-neg-in 97.2%

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

      12. add-sqr-sqrt 56.0%

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

      13. sqrt-unprod 91.2%

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

      14. sqr-neg 91.2%

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

      15. sqrt-unprod 41.3%

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

      16. add-sqr-sqrt 97.4%

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

      17. sub-neg 97.4%

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

   8. Applied egg-rr 97.4%

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

      1. associate-*r/ 97.5%

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

      2. *-rgt-identity 97.5%

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

   10. Simplified 97.5%

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

   11. Taylor expanded in t1 around inf 45.6%

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

   if -2.1e183 < u < 1.7999999999999998e110

   1. Initial program 68.2%

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

      1. associate-*l/ 77.0%

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

      2. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in t1 around inf 67.5%

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

      1. associate-*r/ 67.5%

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

      2. neg-mul-1 67.5%

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

   6. Simplified 67.5%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+183}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 9: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.3 \cdot 10^{+184}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 6.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.3e+184) (/ (- v) u) (if (<= u 6.8e+110) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.3e+184) {
		tmp = -v / u;
	} else if (u <= 6.8e+110) {
		tmp = -v / t1;
	} else {
		tmp = v / 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 (u <= (-3.3d+184)) then
        tmp = -v / u
    else if (u <= 6.8d+110) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.3e+184) {
		tmp = -v / u;
	} else if (u <= 6.8e+110) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.3e+184:
		tmp = -v / u
	elif u <= 6.8e+110:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.3e+184)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 6.8e+110)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.3e+184)
		tmp = -v / u;
	elseif (u <= 6.8e+110)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.3e+184], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 6.8e+110], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.2999999999999998e184

   1. Initial program 86.1%

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

      1. associate-/r* 96.7%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 96.7%

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

      1. mul-1-neg 96.7%

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

      2. associate-/l* 99.9%

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

      3. distribute-neg-frac 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in t1 around inf 41.9%

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

      1. neg-mul-1 41.9%

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

      2. distribute-neg-frac 41.9%

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

   9. Simplified 41.9%

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

   if -3.2999999999999998e184 < u < 6.8000000000000003e110

   1. Initial program 68.2%

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

      1. associate-*l/ 77.0%

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

      2. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in t1 around inf 67.5%

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

      1. associate-*r/ 67.5%

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

      2. neg-mul-1 67.5%

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

   6. Simplified 67.5%

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

   if 6.8000000000000003e110 < u

   1. Initial program 75.5%

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

      1. associate-/r* 88.9%

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

      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 88.8%

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

      1. mul-1-neg 88.8%

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

      2. associate-/l* 95.4%

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

      3. distribute-neg-frac 95.4%

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

   6. Simplified 95.4%

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

      1. frac-2neg 95.4%

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

      2. distribute-frac-neg 95.4%

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

      3. associate-/l* 88.8%

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

      4. mul-1-neg 88.8%

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

      5. div-inv 88.8%

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

      6. mul-1-neg 88.8%

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

      7. remove-double-neg 88.8%

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

      8. div-inv 88.7%

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

      9. associate-*l* 95.3%

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

      10. div-inv 95.4%

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

      11. distribute-neg-in 95.4%

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

      12. add-sqr-sqrt 56.7%

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

      13. sqrt-unprod 89.3%

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

      14. sqr-neg 89.3%

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

      15. sqrt-unprod 38.6%

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

      16. add-sqr-sqrt 95.7%

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

      17. sub-neg 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. associate-*r/ 95.8%

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

      2. *-rgt-identity 95.8%

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

   10. Simplified 95.8%

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

   11. Taylor expanded in t1 around inf 48.3%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.3 \cdot 10^{+184}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 6.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 10: 23.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.32 \cdot 10^{+98}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.3e+137) (/ v t1) (if (<= t1 1.32e+98) (/ v u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.3e+137) {
		tmp = v / t1;
	} else if (t1 <= 1.32e+98) {
		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 (t1 <= (-4.3d+137)) then
        tmp = v / t1
    else if (t1 <= 1.32d+98) 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 (t1 <= -4.3e+137) {
		tmp = v / t1;
	} else if (t1 <= 1.32e+98) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4.3e+137:
		tmp = v / t1
	elif t1 <= 1.32e+98:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.3e+137)
		tmp = Float64(v / t1);
	elseif (t1 <= 1.32e+98)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -4.3e+137)
		tmp = v / t1;
	elseif (t1 <= 1.32e+98)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.3e+137], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 1.32e+98], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.29999999999999965e137 or 1.3200000000000001e98 < t1

   1. Initial program 48.0%

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

      1. associate-*l/ 53.6%

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

      2. *-commutative 53.6%

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

   3. Simplified 53.6%

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

   4. Taylor expanded in t1 around inf 89.2%

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

      1. expm1-log1p-u 79.9%

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

      2. expm1-udef 52.3%

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

      3. frac-2neg 52.3%

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

      4. metadata-eval 52.3%

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

      5. un-div-inv 52.3%

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

      6. add-sqr-sqrt 24.0%

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

      7. sqrt-unprod 43.6%

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

      8. sqr-neg 43.6%

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

      9. sqrt-unprod 24.4%

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

      10. add-sqr-sqrt 43.2%

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

   6. Applied egg-rr 43.2%

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

      1. expm1-def 40.2%

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

      2. expm1-log1p 40.3%

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

   8. Simplified 40.3%

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

   if -4.29999999999999965e137 < t1 < 1.3200000000000001e98

   1. Initial program 83.2%

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

      1. associate-/r* 89.3%

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

      2. associate-/l* 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in t1 around 0 64.8%

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

      1. mul-1-neg 64.8%

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

      2. associate-/l* 66.5%

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

      3. distribute-neg-frac 66.5%

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

   6. Simplified 66.5%

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

      1. frac-2neg 66.5%

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

      2. distribute-frac-neg 66.5%

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

      3. associate-/l* 64.8%

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

      4. mul-1-neg 64.8%

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

      5. div-inv 64.8%

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

      6. mul-1-neg 64.8%

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

      7. remove-double-neg 64.8%

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

      8. div-inv 64.7%

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

      9. associate-*l* 66.4%

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

      10. div-inv 66.5%

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

      11. distribute-neg-in 66.5%

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

      12. add-sqr-sqrt 33.5%

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

      13. sqrt-unprod 66.9%

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

      14. sqr-neg 66.9%

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

      15. sqrt-unprod 32.8%

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

      16. add-sqr-sqrt 66.2%

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

      17. sub-neg 66.2%

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

   8. Applied egg-rr 66.2%

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

      1. associate-*r/ 66.2%

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

      2. *-rgt-identity 66.2%

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

   10. Simplified 66.2%

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

   11. Taylor expanded in t1 around inf 18.6%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.32 \cdot 10^{+98}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 11: 60.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return -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 = -v / (t1 + u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. associate-/r* 81.5%

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

   2. associate-/l* 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in t1 around inf 61.3%

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

   1. neg-mul-1 61.3%

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

6. Simplified 61.3%

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

7. Final simplification 61.3%

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

Alternative 12: 13.8% 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 71.6%

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

   1. associate-*l/ 77.8%

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

   2. *-commutative 77.8%

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

3. Simplified 77.8%

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

4. Taylor expanded in t1 around inf 53.4%

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

   1. expm1-log1p-u 43.1%

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

   2. expm1-udef 31.3%

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

   3. frac-2neg 31.3%

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

   4. metadata-eval 31.3%

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

   5. un-div-inv 31.3%

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

   6. add-sqr-sqrt 14.3%

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

   7. sqrt-unprod 24.3%

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

   8. sqr-neg 24.3%

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

   9. sqrt-unprod 12.1%

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

   10. add-sqr-sqrt 22.4%

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

6. Applied egg-rr 22.4%

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

   1. expm1-def 14.7%

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

   2. expm1-log1p 15.1%

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

8. Simplified 15.1%

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

9. Final simplification 15.1%

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

Reproduce

herbie shell --seed 2023283 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))