Rosa's DopplerBench

Percentage Accurate: 73.3% → 97.9%

Time: 17.3s

Alternatives: 15

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: 73.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. times-frac 98.8%

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

   2. distribute-frac-neg 98.8%

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

   3. distribute-neg-frac2 98.8%

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

   4. +-commutative 98.8%

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

   5. distribute-neg-in 98.8%

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

   6. unsub-neg 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

   \[\leadsto \frac{-v}{t1 + u} \cdot \frac{t1}{t1 + u} \]
6. Add Preprocessing

Alternative 2: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -1.25 \cdot 10^{-109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.05 \cdot 10^{-116}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(u \cdot \frac{v}{t1}\right) - v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- v) (/ t1 (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -4.7e+141)
     (/ (- v) t1)
     (if (<= t1 -1.25e-109)
       t_1
       (if (<= t1 2.05e-116)
         (/ t1 (* u (/ u (- v))))
         (if (<= t1 6.2e+151) t_1 (/ (- (* 2.0 (* u (/ v t1))) v) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v * (t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -4.7e+141) {
		tmp = -v / t1;
	} else if (t1 <= -1.25e-109) {
		tmp = t_1;
	} else if (t1 <= 2.05e-116) {
		tmp = t1 / (u * (u / -v));
	} else if (t1 <= 6.2e+151) {
		tmp = t_1;
	} else {
		tmp = ((2.0 * (u * (v / t1))) - 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) :: t_1
    real(8) :: tmp
    t_1 = -v * (t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-4.7d+141)) then
        tmp = -v / t1
    else if (t1 <= (-1.25d-109)) then
        tmp = t_1
    else if (t1 <= 2.05d-116) then
        tmp = t1 / (u * (u / -v))
    else if (t1 <= 6.2d+151) then
        tmp = t_1
    else
        tmp = ((2.0d0 * (u * (v / t1))) - v) / t1
    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 <= -4.7e+141) {
		tmp = -v / t1;
	} else if (t1 <= -1.25e-109) {
		tmp = t_1;
	} else if (t1 <= 2.05e-116) {
		tmp = t1 / (u * (u / -v));
	} else if (t1 <= 6.2e+151) {
		tmp = t_1;
	} else {
		tmp = ((2.0 * (u * (v / t1))) - v) / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v * (t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -4.7e+141:
		tmp = -v / t1
	elif t1 <= -1.25e-109:
		tmp = t_1
	elif t1 <= 2.05e-116:
		tmp = t1 / (u * (u / -v))
	elif t1 <= 6.2e+151:
		tmp = t_1
	else:
		tmp = ((2.0 * (u * (v / t1))) - v) / t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) * Float64(t1 / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -4.7e+141)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= -1.25e-109)
		tmp = t_1;
	elseif (t1 <= 2.05e-116)
		tmp = Float64(t1 / Float64(u * Float64(u / Float64(-v))));
	elseif (t1 <= 6.2e+151)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(u * Float64(v / t1))) - v) / t1);
	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 <= -4.7e+141)
		tmp = -v / t1;
	elseif (t1 <= -1.25e-109)
		tmp = t_1;
	elseif (t1 <= 2.05e-116)
		tmp = t1 / (u * (u / -v));
	elseif (t1 <= 6.2e+151)
		tmp = t_1;
	else
		tmp = ((2.0 * (u * (v / t1))) - v) / t1;
	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, -4.7e+141], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, -1.25e-109], t$95$1, If[LessEqual[t1, 2.05e-116], N[(t1 / N[(u * N[(u / (-v)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6.2e+151], t$95$1, N[(N[(N[(2.0 * N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -4.69999999999999979e141

   1. Initial program 54.9%

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

      1. associate-*l/ 55.9%

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

      2. *-commutative 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in t1 around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. neg-mul-1 91.4%

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

   7. Simplified 91.4%

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

   if -4.69999999999999979e141 < t1 < -1.25000000000000005e-109 or 2.0499999999999999e-116 < t1 < 6.2000000000000004e151

   1. Initial program 84.7%

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

      1. associate-*l/ 92.2%

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

      2. *-commutative 92.2%

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

   3. Simplified 92.2%

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

   if -1.25000000000000005e-109 < t1 < 2.0499999999999999e-116

   1. Initial program 83.1%

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

      1. times-frac 97.1%

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

      2. distribute-frac-neg 97.1%

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

      3. distribute-neg-frac2 97.1%

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

      4. +-commutative 97.1%

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

      5. distribute-neg-in 97.1%

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

      6. unsub-neg 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t1 around 0 90.0%

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

      1. *-commutative 90.0%

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

      2. clear-num 89.9%

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

      3. frac-2neg 89.9%

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

      4. frac-times 90.7%

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

      5. *-un-lft-identity 90.7%

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

      6. neg-sub0 90.7%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \color{blue}{\left(0 - \left(\left(-u\right) - t1\right)\right)}} \]

      7. add-sqr-sqrt 48.3%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} - t1\right)\right)} \]

      8. sqrt-unprod 76.8%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} - t1\right)\right)} \]

      9. sqr-neg 76.8%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\sqrt{\color{blue}{u \cdot u}} - t1\right)\right)} \]

      10. sqrt-unprod 30.4%

          \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{\sqrt{u} \cdot \sqrt{u}} - t1\right)\right)} \]

      11. add-sqr-sqrt 59.9%

          \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{u} - t1\right)\right)} \]

      12. associate-+l- 59.9%

          \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \color{blue}{\left(\left(0 - u\right) + t1\right)}} \]

      13. neg-sub0 59.9%

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

      14. add-sqr-sqrt 29.6%

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

      15. sqrt-unprod 71.0%

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

      16. sqr-neg 71.0%

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

      17. sqrt-unprod 42.3%

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

      18. add-sqr-sqrt 90.7%

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

      19. +-commutative 90.7%

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

   7. Applied egg-rr 90.7%

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

   8. Taylor expanded in t1 around 0 95.0%

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

   if 6.2000000000000004e151 < t1

   1. Initial program 38.8%

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

      1. associate-*l/ 40.5%

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

      2. *-commutative 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. +-commutative 85.4%

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

      2. neg-mul-1 85.4%

         \[\leadsto \frac{2 \cdot \frac{u \cdot v}{t1} + \color{blue}{\left(-v\right)}}{t1} \]

      3. unsub-neg 85.4%

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

      4. associate-/l* 89.5%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(u \cdot \frac{v}{t1}\right)} - v}{t1} \]

   7. Simplified 89.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(u \cdot \frac{v}{t1}\right) - v}{t1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -1.25 \cdot 10^{-109}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.05 \cdot 10^{-116}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{+151}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(u \cdot \frac{v}{t1}\right) - v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;t1 \leq 2.7 \cdot 10^{+144}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(u \cdot \frac{v}{t1}\right) - v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.5e+152)
   (/ (- v) (+ t1 (* u 2.0)))
   (if (<= t1 2.7e+144)
     (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))
     (/ (- (* 2.0 (* u (/ v t1))) v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.5e+152) {
		tmp = -v / (t1 + (u * 2.0));
	} else if (t1 <= 2.7e+144) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = ((2.0 * (u * (v / t1))) - 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 <= (-1.5d+152)) then
        tmp = -v / (t1 + (u * 2.0d0))
    else if (t1 <= 2.7d+144) then
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    else
        tmp = ((2.0d0 * (u * (v / t1))) - v) / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.5e+152) {
		tmp = -v / (t1 + (u * 2.0));
	} else if (t1 <= 2.7e+144) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = ((2.0 * (u * (v / t1))) - v) / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.5e+152:
		tmp = -v / (t1 + (u * 2.0))
	elif t1 <= 2.7e+144:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	else:
		tmp = ((2.0 * (u * (v / t1))) - v) / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.5e+152)
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	elseif (t1 <= 2.7e+144)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(u * Float64(v / t1))) - v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.5e+152)
		tmp = -v / (t1 + (u * 2.0));
	elseif (t1 <= 2.7e+144)
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	else
		tmp = ((2.0 * (u * (v / t1))) - v) / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.5e+152], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.7e+144], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.49999999999999995e152

   1. Initial program 53.9%

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

      1. associate-*l/ 54.9%

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

      2. *-commutative 54.9%

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

   3. Simplified 54.9%

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

      1. associate-*r/ 53.9%

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

      2. times-frac 100.0%

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

      3. *-commutative 100.0%

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

      4. frac-2neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. sub-neg 100.0%

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

      9. clear-num 100.0%

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

      10. frac-2neg 100.0%

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

      11. frac-times 100.0%

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

      12. *-un-lft-identity 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-neg-in 100.0%

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

      15. sub-neg 100.0%

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

      16. sub-neg 100.0%

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

      17. +-commutative 100.0%

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

      18. add-sqr-sqrt 99.5%

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

      19. sqrt-unprod 54.9%

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

      20. sqr-neg 54.9%

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

      21. sqrt-unprod 0.0%

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

      22. add-sqr-sqrt 0.0%

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

      23. sqrt-unprod 0.0%

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

      24. sqr-neg 0.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in u around 0 96.7%

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

      1. *-commutative 96.7%

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

   9. Simplified 96.7%

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

   if -1.49999999999999995e152 < t1 < 2.70000000000000015e144

   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-/l* 88.7%

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

   3. Simplified 88.7%

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

      1. associate-/r* 95.3%

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

      2. div-inv 95.3%

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

   6. Applied egg-rr 95.3%

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

      1. associate-*r/ 95.3%

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

      2. *-rgt-identity 95.3%

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

   8. Simplified 95.3%

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

   if 2.70000000000000015e144 < t1

   1. Initial program 38.8%

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

      1. associate-*l/ 40.5%

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

      2. *-commutative 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. +-commutative 85.4%

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

      2. neg-mul-1 85.4%

         \[\leadsto \frac{2 \cdot \frac{u \cdot v}{t1} + \color{blue}{\left(-v\right)}}{t1} \]

      3. unsub-neg 85.4%

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

      4. associate-/l* 89.5%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(u \cdot \frac{v}{t1}\right)} - v}{t1} \]

   7. Simplified 89.5%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;t1 \leq 2.7 \cdot 10^{+144}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(u \cdot \frac{v}{t1}\right) - v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{+143}:\\ \;\;\;\;t1 \cdot \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(u \cdot \frac{v}{t1}\right) - v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.9e+131)
   (/ (- v) t1)
   (if (<= t1 4.8e+143)
     (* t1 (/ (- v) (* (+ t1 u) (+ t1 u))))
     (/ (- (* 2.0 (* u (/ v t1))) v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.9e+131) {
		tmp = -v / t1;
	} else if (t1 <= 4.8e+143) {
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)));
	} else {
		tmp = ((2.0 * (u * (v / t1))) - 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 <= (-2.9d+131)) then
        tmp = -v / t1
    else if (t1 <= 4.8d+143) then
        tmp = t1 * (-v / ((t1 + u) * (t1 + u)))
    else
        tmp = ((2.0d0 * (u * (v / t1))) - v) / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.9e+131) {
		tmp = -v / t1;
	} else if (t1 <= 4.8e+143) {
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)));
	} else {
		tmp = ((2.0 * (u * (v / t1))) - v) / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.9e+131:
		tmp = -v / t1
	elif t1 <= 4.8e+143:
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)))
	else:
		tmp = ((2.0 * (u * (v / t1))) - v) / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.9e+131)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 4.8e+143)
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(u * Float64(v / t1))) - v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.9e+131)
		tmp = -v / t1;
	elseif (t1 <= 4.8e+143)
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)));
	else
		tmp = ((2.0 * (u * (v / t1))) - v) / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.9e+131], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 4.8e+143], N[(t1 * N[((-v) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.9000000000000001e131

   1. Initial program 54.9%

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

      1. associate-*l/ 55.9%

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

      2. *-commutative 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in t1 around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. neg-mul-1 91.4%

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

   7. Simplified 91.4%

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

   if -2.9000000000000001e131 < t1 < 4.79999999999999959e143

   1. Initial program 83.9%

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

      1. associate-/l* 89.9%

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

   3. Simplified 89.9%

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

   if 4.79999999999999959e143 < t1

   1. Initial program 38.8%

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

      1. associate-*l/ 40.5%

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

      2. *-commutative 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. +-commutative 85.4%

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

      2. neg-mul-1 85.4%

         \[\leadsto \frac{2 \cdot \frac{u \cdot v}{t1} + \color{blue}{\left(-v\right)}}{t1} \]

      3. unsub-neg 85.4%

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

      4. associate-/l* 89.5%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(u \cdot \frac{v}{t1}\right)} - v}{t1} \]

   7. Simplified 89.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{+143}:\\ \;\;\;\;t1 \cdot \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(u \cdot \frac{v}{t1}\right) - v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.05 \cdot 10^{+21}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 9.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.05e+21)
   (/ (- v) t1)
   (if (<= t1 9.2e-29) (/ t1 (* u (/ u (- v)))) (/ (- v) (+ t1 (* u 2.0))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.05e+21) {
		tmp = -v / t1;
	} else if (t1 <= 9.2e-29) {
		tmp = t1 / (u * (u / -v));
	} else {
		tmp = -v / (t1 + (u * 2.0));
	}
	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 <= (-2.05d+21)) then
        tmp = -v / t1
    else if (t1 <= 9.2d-29) then
        tmp = t1 / (u * (u / -v))
    else
        tmp = -v / (t1 + (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.05e+21) {
		tmp = -v / t1;
	} else if (t1 <= 9.2e-29) {
		tmp = t1 / (u * (u / -v));
	} else {
		tmp = -v / (t1 + (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.05e+21:
		tmp = -v / t1
	elif t1 <= 9.2e-29:
		tmp = t1 / (u * (u / -v))
	else:
		tmp = -v / (t1 + (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.05e+21)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 9.2e-29)
		tmp = Float64(t1 / Float64(u * Float64(u / Float64(-v))));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.05e+21)
		tmp = -v / t1;
	elseif (t1 <= 9.2e-29)
		tmp = t1 / (u * (u / -v));
	else
		tmp = -v / (t1 + (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.05e+21], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 9.2e-29], N[(t1 / N[(u * N[(u / (-v)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.05e21

   1. Initial program 61.7%

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

      1. associate-*l/ 69.2%

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

      2. *-commutative 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in t1 around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. neg-mul-1 87.9%

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

   7. Simplified 87.9%

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

   if -2.05e21 < t1 < 9.19999999999999965e-29

   1. Initial program 86.5%

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

      1. times-frac 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. distribute-neg-frac2 98.0%

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

      4. +-commutative 98.0%

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

      5. distribute-neg-in 98.0%

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

      6. unsub-neg 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around 0 82.5%

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

      1. *-commutative 82.5%

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

      2. clear-num 82.5%

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

      3. frac-2neg 82.5%

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

      4. frac-times 83.1%

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

      5. *-un-lft-identity 83.1%

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

      6. neg-sub0 83.1%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \color{blue}{\left(0 - \left(\left(-u\right) - t1\right)\right)}} \]

      7. add-sqr-sqrt 46.7%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} - t1\right)\right)} \]

      8. sqrt-unprod 70.5%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} - t1\right)\right)} \]

      9. sqr-neg 70.5%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\sqrt{\color{blue}{u \cdot u}} - t1\right)\right)} \]

      10. sqrt-unprod 25.7%

          \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{\sqrt{u} \cdot \sqrt{u}} - t1\right)\right)} \]

      11. add-sqr-sqrt 54.5%

          \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{u} - t1\right)\right)} \]

      12. associate-+l- 54.5%

          \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \color{blue}{\left(\left(0 - u\right) + t1\right)}} \]

      13. neg-sub0 54.5%

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

      14. add-sqr-sqrt 28.9%

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

      15. sqrt-unprod 64.5%

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

      16. sqr-neg 64.5%

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

      17. sqrt-unprod 36.2%

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

      18. add-sqr-sqrt 83.1%

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

      19. +-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

   8. Taylor expanded in t1 around 0 85.9%

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

   if 9.19999999999999965e-29 < t1

   1. Initial program 62.6%

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

      1. associate-*l/ 66.6%

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

      2. *-commutative 66.6%

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

   3. Simplified 66.6%

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

      1. associate-*r/ 62.6%

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

      2. times-frac 99.8%

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

      3. *-commutative 99.8%

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

      4. frac-2neg 99.8%

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

      5. remove-double-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-neg-in 99.8%

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

      8. sub-neg 99.8%

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

      9. clear-num 99.8%

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

      10. frac-2neg 99.8%

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

      11. frac-times 95.2%

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

      12. *-un-lft-identity 95.2%

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

      13. +-commutative 95.2%

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

      14. distribute-neg-in 95.2%

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

      15. sub-neg 95.2%

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

      16. sub-neg 95.2%

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

      17. +-commutative 95.2%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 36.7%

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

      20. sqr-neg 36.7%

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

      21. sqrt-unprod 38.6%

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

      22. add-sqr-sqrt 15.7%

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

      23. sqrt-unprod 26.6%

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

      24. sqr-neg 26.6%

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in u around 0 77.8%

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

      1. *-commutative 77.8%

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

   9. Simplified 77.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.05 \cdot 10^{+21}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 9.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{-29}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -3.5e+24)
   (/ (- v) t1)
   (if (<= t1 9e-29) (/ t1 (* u (/ u (- v)))) (/ (- v) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.5e+24) {
		tmp = -v / t1;
	} else if (t1 <= 9e-29) {
		tmp = t1 / (u * (u / -v));
	} 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 <= (-3.5d+24)) then
        tmp = -v / t1
    else if (t1 <= 9d-29) then
        tmp = t1 / (u * (u / -v))
    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 <= -3.5e+24) {
		tmp = -v / t1;
	} else if (t1 <= 9e-29) {
		tmp = t1 / (u * (u / -v));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -3.5e+24:
		tmp = -v / t1
	elif t1 <= 9e-29:
		tmp = t1 / (u * (u / -v))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -3.5e+24)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 9e-29)
		tmp = Float64(t1 / Float64(u * Float64(u / Float64(-v))));
	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 <= -3.5e+24)
		tmp = -v / t1;
	elseif (t1 <= 9e-29)
		tmp = t1 / (u * (u / -v));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -3.5e+24], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 9e-29], N[(t1 / N[(u * N[(u / (-v)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.5000000000000002e24

   1. Initial program 61.7%

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

      1. associate-*l/ 69.2%

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

      2. *-commutative 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in t1 around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. neg-mul-1 87.9%

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

   7. Simplified 87.9%

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

   if -3.5000000000000002e24 < t1 < 8.9999999999999996e-29

   1. Initial program 86.5%

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

      1. times-frac 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. distribute-neg-frac2 98.0%

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

      4. +-commutative 98.0%

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

      5. distribute-neg-in 98.0%

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

      6. unsub-neg 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around 0 82.5%

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

      1. *-commutative 82.5%

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

      2. clear-num 82.5%

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

      3. frac-2neg 82.5%

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

      4. frac-times 83.1%

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

      5. *-un-lft-identity 83.1%

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

      6. neg-sub0 83.1%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \color{blue}{\left(0 - \left(\left(-u\right) - t1\right)\right)}} \]

      7. add-sqr-sqrt 46.7%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} - t1\right)\right)} \]

      8. sqrt-unprod 70.5%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} - t1\right)\right)} \]

      9. sqr-neg 70.5%

         \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\sqrt{\color{blue}{u \cdot u}} - t1\right)\right)} \]

      10. sqrt-unprod 25.7%

          \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{\sqrt{u} \cdot \sqrt{u}} - t1\right)\right)} \]

      11. add-sqr-sqrt 54.5%

          \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \left(0 - \left(\color{blue}{u} - t1\right)\right)} \]

      12. associate-+l- 54.5%

          \[\leadsto \frac{-t1}{\frac{u}{v} \cdot \color{blue}{\left(\left(0 - u\right) + t1\right)}} \]

      13. neg-sub0 54.5%

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

      14. add-sqr-sqrt 28.9%

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

      15. sqrt-unprod 64.5%

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

      16. sqr-neg 64.5%

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

      17. sqrt-unprod 36.2%

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

      18. add-sqr-sqrt 83.1%

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

      19. +-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

   8. Taylor expanded in t1 around 0 85.9%

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

   if 8.9999999999999996e-29 < t1

   1. Initial program 62.6%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 77.7%

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

      1. mul-1-neg 77.7%

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

      2. neg-sub0 77.7%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

      3. +-commutative 77.7%

         \[\leadsto 0 - \frac{v}{\color{blue}{u + t1}} \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{0 - \frac{v}{u + t1}} \]
   8. Step-by-step derivation

      1. neg-sub0 77.7%

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

      2. distribute-neg-frac 77.7%

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

   9. Simplified 77.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{-29}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-29}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -3e+25)
   (/ (- v) t1)
   (if (<= t1 8.5e-29) (* t1 (/ (/ v u) (- u))) (/ (- v) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3e+25) {
		tmp = -v / t1;
	} else if (t1 <= 8.5e-29) {
		tmp = t1 * ((v / 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 <= (-3d+25)) then
        tmp = -v / t1
    else if (t1 <= 8.5d-29) then
        tmp = t1 * ((v / 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 <= -3e+25) {
		tmp = -v / t1;
	} else if (t1 <= 8.5e-29) {
		tmp = t1 * ((v / u) / -u);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -3e+25:
		tmp = -v / t1
	elif t1 <= 8.5e-29:
		tmp = t1 * ((v / u) / -u)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -3e+25)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 8.5e-29)
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-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 <= -3e+25)
		tmp = -v / t1;
	elseif (t1 <= 8.5e-29)
		tmp = t1 * ((v / u) / -u);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -3e+25], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 8.5e-29], N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.00000000000000006e25

   1. Initial program 61.7%

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

      1. associate-*l/ 69.2%

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

      2. *-commutative 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in t1 around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. neg-mul-1 87.9%

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

   7. Simplified 87.9%

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

   if -3.00000000000000006e25 < t1 < 8.5000000000000001e-29

   1. Initial program 86.5%

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

      1. times-frac 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. distribute-neg-frac2 98.0%

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

      4. +-commutative 98.0%

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

      5. distribute-neg-in 98.0%

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

      6. unsub-neg 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around 0 82.5%

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

   6. Taylor expanded in v around 0 73.1%

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

      1. mul-1-neg 73.1%

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

      2. associate-/l* 79.1%

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

      3. distribute-rgt-neg-in 79.1%

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

   8. Simplified 79.1%

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

   9. Taylor expanded in t1 around 0 81.9%

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

       1. associate-/r* 85.9%

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

       2. distribute-neg-frac2 85.9%

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

   11. Applied egg-rr 85.9%

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

   if 8.5000000000000001e-29 < t1

   1. Initial program 62.6%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 77.7%

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

      1. mul-1-neg 77.7%

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

      2. neg-sub0 77.7%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

      3. +-commutative 77.7%

         \[\leadsto 0 - \frac{v}{\color{blue}{u + t1}} \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{0 - \frac{v}{u + t1}} \]
   8. Step-by-step derivation

      1. neg-sub0 77.7%

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

      2. distribute-neg-frac 77.7%

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

   9. Simplified 77.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-29}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.75 \cdot 10^{+25}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{-29}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -3.75e+25)
   (/ (- v) t1)
   (if (<= t1 9e-29) (* t1 (/ (- v) (* u u))) (/ (- v) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.75e+25) {
		tmp = -v / t1;
	} else if (t1 <= 9e-29) {
		tmp = t1 * (-v / (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 <= (-3.75d+25)) then
        tmp = -v / t1
    else if (t1 <= 9d-29) then
        tmp = t1 * (-v / (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 <= -3.75e+25) {
		tmp = -v / t1;
	} else if (t1 <= 9e-29) {
		tmp = t1 * (-v / (u * u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -3.75e+25:
		tmp = -v / t1
	elif t1 <= 9e-29:
		tmp = t1 * (-v / (u * u))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -3.75e+25)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 9e-29)
		tmp = Float64(t1 * Float64(Float64(-v) / 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 <= -3.75e+25)
		tmp = -v / t1;
	elseif (t1 <= 9e-29)
		tmp = t1 * (-v / (u * u));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -3.75e+25], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 9e-29], N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.74999999999999996e25

   1. Initial program 61.7%

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

      1. associate-*l/ 69.2%

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

      2. *-commutative 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in t1 around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. neg-mul-1 87.9%

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

   7. Simplified 87.9%

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

   if -3.74999999999999996e25 < t1 < 8.9999999999999996e-29

   1. Initial program 86.5%

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

      1. times-frac 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. distribute-neg-frac2 98.0%

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

      4. +-commutative 98.0%

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

      5. distribute-neg-in 98.0%

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

      6. unsub-neg 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around 0 82.5%

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

   6. Taylor expanded in v around 0 73.1%

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

      1. mul-1-neg 73.1%

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

      2. associate-/l* 79.1%

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

      3. distribute-rgt-neg-in 79.1%

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

   8. Simplified 79.1%

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

   9. Taylor expanded in t1 around 0 81.9%

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

   if 8.9999999999999996e-29 < t1

   1. Initial program 62.6%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 77.7%

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

      1. mul-1-neg 77.7%

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

      2. neg-sub0 77.7%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

      3. +-commutative 77.7%

         \[\leadsto 0 - \frac{v}{\color{blue}{u + t1}} \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{0 - \frac{v}{u + t1}} \]
   8. Step-by-step derivation

      1. neg-sub0 77.7%

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

      2. distribute-neg-frac 77.7%

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

   9. Simplified 77.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.75 \cdot 10^{+25}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{-29}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.85 \cdot 10^{+84} \lor \neg \left(u \leq 0.25\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.85e+84) (not (<= u 0.25)))
   (* t1 (/ (/ v u) u))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.85e+84) || !(u <= 0.25)) {
		tmp = t1 * ((v / 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 ((u <= (-1.85d+84)) .or. (.not. (u <= 0.25d0))) then
        tmp = t1 * ((v / 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 ((u <= -1.85e+84) || !(u <= 0.25)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.85e+84) or not (u <= 0.25):
		tmp = t1 * ((v / u) / u)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.85e+84) || !(u <= 0.25))
		tmp = Float64(t1 * Float64(Float64(v / 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 ((u <= -1.85e+84) || ~((u <= 0.25)))
		tmp = t1 * ((v / u) / u);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.85e+84], N[Not[LessEqual[u, 0.25]], $MachinePrecision]], N[(t1 * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.85e84 or 0.25 < u

   1. Initial program 81.5%

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

      1. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around 0 87.6%

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

   6. Taylor expanded in v around 0 79.3%

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

      1. mul-1-neg 79.3%

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

      2. associate-/l* 81.5%

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

      3. distribute-rgt-neg-in 81.5%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in t1 around 0 78.1%

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

       1. add-sqr-sqrt 66.3%

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

       2. sqrt-unprod 72.9%

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

       3. sqr-neg 72.9%

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

       4. sqrt-unprod 61.0%

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

       5. add-sqr-sqrt 67.7%

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

       6. associate-/r* 67.5%

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

   11. Applied egg-rr 67.5%

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

   if -1.85e84 < u < 0.25

   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. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. neg-sub0 69.3%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

      3. +-commutative 69.3%

         \[\leadsto 0 - \frac{v}{\color{blue}{u + t1}} \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{0 - \frac{v}{u + t1}} \]
   8. Step-by-step derivation

      1. neg-sub0 69.3%

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

      2. distribute-neg-frac 69.3%

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

   9. Simplified 69.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.85 \cdot 10^{+84} \lor \neg \left(u \leq 0.25\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.95 \cdot 10^{+72} \lor \neg \left(u \leq 4.8 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.95e+72) (not (<= u 4.8e+130))) (/ 1.0 (/ u v)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.95e+72) || !(u <= 4.8e+130)) {
		tmp = 1.0 / (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 <= (-2.95d+72)) .or. (.not. (u <= 4.8d+130))) then
        tmp = 1.0d0 / (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 <= -2.95e+72) || !(u <= 4.8e+130)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.95e+72) or not (u <= 4.8e+130):
		tmp = 1.0 / (u / v)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.95e+72) || !(u <= 4.8e+130))
		tmp = Float64(1.0 / 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 <= -2.95e+72) || ~((u <= 4.8e+130)))
		tmp = 1.0 / (u / v);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.95e+72], N[Not[LessEqual[u, 4.8e+130]], $MachinePrecision]], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.9500000000000001e72 or 4.80000000000000048e130 < u

   1. Initial program 79.2%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around inf 35.6%

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

   6. Taylor expanded in t1 around 0 29.3%

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

      1. associate-*r/ 29.3%

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

      2. mul-1-neg 29.3%

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

   8. Simplified 29.3%

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

      1. add-sqr-sqrt 13.3%

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

      2. sqrt-unprod 31.2%

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

      3. sqr-neg 31.2%

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

      4. sqrt-unprod 16.1%

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

      5. add-sqr-sqrt 29.7%

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

      6. clear-num 31.7%

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

      7. inv-pow 31.7%

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

   10. Applied egg-rr 31.7%

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

       1. unpow-1 31.7%

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

   12. Simplified 31.7%

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

   if -2.9500000000000001e72 < u < 4.80000000000000048e130

   1. Initial program 73.8%

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

      1. associate-*l/ 81.8%

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

      2. *-commutative 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in t1 around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.95 \cdot 10^{+72} \lor \neg \left(u \leq 4.8 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+170} \lor \neg \left(u \leq 7.2 \cdot 10^{+131}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.8e+170) (not (<= u 7.2e+131))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.8e+170) || !(u <= 7.2e+131)) {
		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 <= (-2.8d+170)) .or. (.not. (u <= 7.2d+131))) 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 <= -2.8e+170) || !(u <= 7.2e+131)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.8e+170) or not (u <= 7.2e+131):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.8e+170) || !(u <= 7.2e+131))
		tmp = 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 <= -2.8e+170) || ~((u <= 7.2e+131)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.8e+170], N[Not[LessEqual[u, 7.2e+131]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.80000000000000015e170 or 7.20000000000000063e131 < u

   1. Initial program 77.1%

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

      1. times-frac 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. distribute-neg-frac2 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-neg-in 98.5%

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

      6. unsub-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around inf 34.8%

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

   6. Taylor expanded in t1 around 0 31.9%

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

      1. associate-*r/ 31.9%

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

      2. mul-1-neg 31.9%

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

   8. Simplified 31.9%

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

      1. div-inv 31.9%

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

      2. add-sqr-sqrt 13.5%

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

      3. sqrt-unprod 31.7%

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

      4. sqr-neg 31.7%

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

      5. sqrt-unprod 18.5%

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

      6. add-sqr-sqrt 32.4%

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

   10. Applied egg-rr 32.4%

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

       1. associate-*r/ 32.4%

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

       2. *-rgt-identity 32.4%

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

   12. Simplified 32.4%

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

   if -2.80000000000000015e170 < u < 7.20000000000000063e131

   1. Initial program 75.3%

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

      1. associate-*l/ 80.7%

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

      2. *-commutative 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in t1 around inf 59.3%

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

      1. associate-*r/ 59.3%

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

      2. neg-mul-1 59.3%

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

   7. Simplified 59.3%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+170} \lor \neg \left(u \leq 7.2 \cdot 10^{+131}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 22.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6.5 \cdot 10^{+29} \lor \neg \left(t1 \leq 9.2 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -6.5e+29) (not (<= t1 9.2e+167))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6.5e+29) || !(t1 <= 9.2e+167)) {
		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 ((t1 <= (-6.5d+29)) .or. (.not. (t1 <= 9.2d+167))) 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 ((t1 <= -6.5e+29) || !(t1 <= 9.2e+167)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -6.5e+29) or not (t1 <= 9.2e+167):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -6.5e+29) || !(t1 <= 9.2e+167))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -6.5e+29) || ~((t1 <= 9.2e+167)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -6.5e+29], N[Not[LessEqual[t1, 9.2e+167]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.49999999999999971e29 or 9.19999999999999952e167 < t1

   1. Initial program 52.7%

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

      1. associate-*l/ 58.8%

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

      2. *-commutative 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in t1 around inf 88.8%

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

      1. associate-*r/ 88.8%

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

      2. neg-mul-1 88.8%

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

   7. Simplified 88.8%

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

      1. neg-sub0 88.8%

         \[\leadsto \frac{\color{blue}{0 - v}}{t1} \]

      2. sub-neg 88.8%

         \[\leadsto \frac{\color{blue}{0 + \left(-v\right)}}{t1} \]

      3. add-sqr-sqrt 38.4%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1} \]

      4. sqrt-unprod 54.9%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{t1} \]

      5. sqr-neg 54.9%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{v \cdot v}}}{t1} \]

      6. sqrt-unprod 22.1%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1} \]

      7. add-sqr-sqrt 36.6%

         \[\leadsto \frac{0 + \color{blue}{v}}{t1} \]

   9. Applied egg-rr 36.6%

      \[\leadsto \frac{\color{blue}{0 + v}}{t1} \]
   10. Step-by-step derivation

       1. +-lft-identity 36.6%

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

   11. Simplified 36.6%

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

   if -6.49999999999999971e29 < t1 < 9.19999999999999952e167

   1. Initial program 85.1%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around inf 40.4%

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

   6. Taylor expanded in t1 around 0 16.8%

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

      1. associate-*r/ 16.8%

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

      2. mul-1-neg 16.8%

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

   8. Simplified 16.8%

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

      1. div-inv 16.8%

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

      2. add-sqr-sqrt 7.8%

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

      3. sqrt-unprod 23.9%

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

      4. sqr-neg 23.9%

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

      5. sqrt-unprod 8.6%

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

      6. add-sqr-sqrt 17.5%

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

   10. Applied egg-rr 17.5%

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

       1. associate-*r/ 17.5%

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

       2. *-rgt-identity 17.5%

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

   12. Simplified 17.5%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6.5 \cdot 10^{+29} \lor \neg \left(t1 \leq 9.2 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.05 \cdot 10^{-221}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 1.05e-221) (/ v (+ t1 u)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 1.05e-221) {
		tmp = v / (t1 + 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 (v <= 1.05d-221) then
        tmp = v / (t1 + 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 (v <= 1.05e-221) {
		tmp = v / (t1 + u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 1.05e-221:
		tmp = v / (t1 + u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 1.05e-221)
		tmp = Float64(v / Float64(t1 + u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 1.05e-221)
		tmp = v / (t1 + u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 1.05e-221], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.05e-221

   1. Initial program 77.1%

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

      1. times-frac 98.4%

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

      2. distribute-frac-neg 98.4%

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

      3. distribute-neg-frac2 98.4%

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

      4. +-commutative 98.4%

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

      5. distribute-neg-in 98.4%

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

      6. unsub-neg 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in t1 around inf 56.4%

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

      1. associate-*r/ 56.4%

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

      2. neg-mul-1 56.4%

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

      3. add-sqr-sqrt 45.5%

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

      4. sqrt-unprod 58.2%

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

      5. sqr-neg 58.2%

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

      6. sqrt-unprod 10.8%

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

      7. add-sqr-sqrt 28.0%

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

      8. +-commutative 28.0%

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

   7. Applied egg-rr 28.0%

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

   if 1.05e-221 < v

   1. Initial program 74.2%

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

   5. Taylor expanded in t1 around inf 49.6%

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

      1. associate-*r/ 49.6%

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

      2. neg-mul-1 49.6%

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

   7. Simplified 49.6%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.05 \cdot 10^{-221}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.1% 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 75.8%

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

   1. times-frac 98.8%

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

   2. distribute-frac-neg 98.8%

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

   3. distribute-neg-frac2 98.8%

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

   4. +-commutative 98.8%

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

   5. distribute-neg-in 98.8%

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

   6. unsub-neg 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in t1 around inf 54.3%

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

   1. mul-1-neg 54.3%

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

   2. neg-sub0 54.3%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

   3. +-commutative 54.3%

      \[\leadsto 0 - \frac{v}{\color{blue}{u + t1}} \]

7. Applied egg-rr 54.3%

   \[\leadsto \color{blue}{0 - \frac{v}{u + t1}} \]
8. Step-by-step derivation

   1. neg-sub0 54.3%

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

   2. distribute-neg-frac 54.3%

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

9. Simplified 54.3%

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

10. Final simplification 54.3%

    \[\leadsto \frac{-v}{t1 + u} \]
11. Add Preprocessing

Alternative 15: 14.1% 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 75.8%

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

   1. associate-*l/ 79.8%

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

   2. *-commutative 79.8%

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

3. Simplified 79.8%

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

5. Taylor expanded in t1 around inf 46.1%

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

   1. associate-*r/ 46.1%

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

   2. neg-mul-1 46.1%

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

7. Simplified 46.1%

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

   1. neg-sub0 46.1%

      \[\leadsto \frac{\color{blue}{0 - v}}{t1} \]

   2. sub-neg 46.1%

      \[\leadsto \frac{\color{blue}{0 + \left(-v\right)}}{t1} \]

   3. add-sqr-sqrt 20.6%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1} \]

   4. sqrt-unprod 36.5%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{t1} \]

   5. sqr-neg 36.5%

      \[\leadsto \frac{0 + \sqrt{\color{blue}{v \cdot v}}}{t1} \]

   6. sqrt-unprod 7.2%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1} \]

   7. add-sqr-sqrt 12.2%

      \[\leadsto \frac{0 + \color{blue}{v}}{t1} \]

9. Applied egg-rr 12.2%

   \[\leadsto \frac{\color{blue}{0 + v}}{t1} \]
10. Step-by-step derivation

    1. +-lft-identity 12.2%

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

11. Simplified 12.2%

    \[\leadsto \frac{\color{blue}{v}}{t1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))