Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.1%

Time: 15.4s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% 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: 98.1% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 73.7%

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

   1. associate-/r* 83.0%

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

   2. associate-/l* 97.8%

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

3. Simplified 97.8%

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

   1. div-inv 97.5%

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

   2. clear-num 98.1%

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

   3. add-sqr-sqrt 50.3%

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

   4. sqrt-unprod 46.0%

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

   5. sqr-neg 46.0%

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

   6. sqrt-unprod 18.2%

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

   7. add-sqr-sqrt 35.1%

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

   8. distribute-lft-neg-in 35.1%

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

   9. distribute-rgt-neg-in 35.1%

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

   10. add-sqr-sqrt 18.2%

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

   11. sqrt-unprod 46.0%

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

   12. sqr-neg 46.0%

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

   13. sqrt-unprod 50.3%

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

   14. add-sqr-sqrt 98.1%

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

5. Applied egg-rr 98.1%

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

6. Final simplification 98.1%

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

Alternative 2: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;u \leq -7.4 \cdot 10^{+170}:\\ \;\;\;\;-\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq -4.8 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= u -7.4e+170)
     (- (/ t1 (* u (/ u v))))
     (if (<= u -4.8e-127)
       t_1
       (if (<= u 8.5e-162)
         (/ (- v) t1)
         (if (<= u 7.2e+103) t_1 (/ (/ t1 (+ t1 u)) (/ (- t1 u) v))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (u <= -7.4e+170) {
		tmp = -(t1 / (u * (u / v)));
	} else if (u <= -4.8e-127) {
		tmp = t_1;
	} else if (u <= 8.5e-162) {
		tmp = -v / t1;
	} else if (u <= 7.2e+103) {
		tmp = t_1;
	} else {
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	}
	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 (u <= (-7.4d+170)) then
        tmp = -(t1 / (u * (u / v)))
    else if (u <= (-4.8d-127)) then
        tmp = t_1
    else if (u <= 8.5d-162) then
        tmp = -v / t1
    else if (u <= 7.2d+103) then
        tmp = t_1
    else
        tmp = (t1 / (t1 + u)) / ((t1 - u) / v)
    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 (u <= -7.4e+170) {
		tmp = -(t1 / (u * (u / v)));
	} else if (u <= -4.8e-127) {
		tmp = t_1;
	} else if (u <= 8.5e-162) {
		tmp = -v / t1;
	} else if (u <= 7.2e+103) {
		tmp = t_1;
	} else {
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if u <= -7.4e+170:
		tmp = -(t1 / (u * (u / v)))
	elif u <= -4.8e-127:
		tmp = t_1
	elif u <= 8.5e-162:
		tmp = -v / t1
	elif u <= 7.2e+103:
		tmp = t_1
	else:
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (u <= -7.4e+170)
		tmp = Float64(-Float64(t1 / Float64(u * Float64(u / v))));
	elseif (u <= -4.8e-127)
		tmp = t_1;
	elseif (u <= 8.5e-162)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 7.2e+103)
		tmp = t_1;
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) / Float64(Float64(t1 - u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (u <= -7.4e+170)
		tmp = -(t1 / (u * (u / v)));
	elseif (u <= -4.8e-127)
		tmp = t_1;
	elseif (u <= 8.5e-162)
		tmp = -v / t1;
	elseif (u <= 7.2e+103)
		tmp = t_1;
	else
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	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[u, -7.4e+170], (-N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[u, -4.8e-127], t$95$1, If[LessEqual[u, 8.5e-162], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 7.2e+103], t$95$1, N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -7.39999999999999975e170

   1. Initial program 66.9%

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

      1. associate-/r* 89.1%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 88.8%

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

      1. +-commutative 88.8%

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

   6. Simplified 88.8%

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

   7. Taylor expanded in t1 around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. *-commutative 66.9%

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

      3. unpow2 66.9%

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

      4. times-frac 84.0%

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

      5. associate-*l* 84.0%

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

      6. metadata-eval 84.0%

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

      7. times-frac 84.0%

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

      8. *-rgt-identity 84.0%

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

      9. *-commutative 84.0%

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

      10. neg-mul-1 84.0%

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

   9. Simplified 84.0%

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

       1. clear-num 86.6%

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

       2. frac-2neg 86.6%

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

       3. frac-times 91.7%

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

       4. *-un-lft-identity 91.7%

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

       5. remove-double-neg 91.7%

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

   11. Applied egg-rr 91.7%

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

   if -7.39999999999999975e170 < u < -4.79999999999999964e-127 or 8.49999999999999955e-162 < u < 7.20000000000000033e103

   1. Initial program 77.9%

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

      1. associate-*l/ 88.2%

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

      2. *-commutative 88.2%

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

   3. Simplified 88.2%

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

   if -4.79999999999999964e-127 < u < 8.49999999999999955e-162

   1. Initial program 68.3%

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

      1. associate-*l/ 71.5%

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

      2. *-commutative 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in t1 around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. neg-mul-1 91.3%

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

   6. Simplified 91.3%

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

   if 7.20000000000000033e103 < u

   1. Initial program 78.4%

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

      1. associate-*l/ 73.6%

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

      2. *-commutative 73.6%

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

   3. Simplified 73.6%

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

      1. associate-/r* 80.5%

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

      2. associate-*r/ 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-/r/ 98.6%

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

      5. div-inv 98.7%

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

      6. clear-num 98.6%

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

      7. associate-/r/ 98.6%

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

      8. clear-num 99.8%

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

      9. *-commutative 99.8%

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

      10. clear-num 98.6%

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

      11. div-inv 98.7%

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

      12. frac-2neg 98.7%

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

      13. remove-double-neg 98.7%

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

      14. associate-*l/ 97.6%

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

   5. Applied egg-rr 95.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.4 \cdot 10^{+170}:\\ \;\;\;\;-\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq -4.8 \cdot 10^{-127}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1 - u}{v}}\\ \end{array} \]

Alternative 3: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1 + u}\\ t_2 := t1 \cdot \frac{-\frac{v}{u}}{u}\\ \mathbf{if}\;t1 \leq -125000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -5.8 \cdot 10^{-47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -8 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -8.2 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (+ t1 u)))) (t_2 (* t1 (/ (- (/ v u)) u))))
   (if (<= t1 -125000000000.0)
     t_1
     (if (<= t1 -5.8e-47)
       t_2
       (if (<= t1 -8e-77)
         t_1
         (if (<= t1 -8.2e-160)
           t_2
           (if (<= t1 3.4e-17) (* v (/ (/ t1 (- u)) u)) t_1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / (t1 + u));
	double t_2 = t1 * (-(v / u) / u);
	double tmp;
	if (t1 <= -125000000000.0) {
		tmp = t_1;
	} else if (t1 <= -5.8e-47) {
		tmp = t_2;
	} else if (t1 <= -8e-77) {
		tmp = t_1;
	} else if (t1 <= -8.2e-160) {
		tmp = t_2;
	} else if (t1 <= 3.4e-17) {
		tmp = v * ((t1 / -u) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -(v / (t1 + u))
    t_2 = t1 * (-(v / u) / u)
    if (t1 <= (-125000000000.0d0)) then
        tmp = t_1
    else if (t1 <= (-5.8d-47)) then
        tmp = t_2
    else if (t1 <= (-8d-77)) then
        tmp = t_1
    else if (t1 <= (-8.2d-160)) then
        tmp = t_2
    else if (t1 <= 3.4d-17) then
        tmp = v * ((t1 / -u) / u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -(v / (t1 + u));
	double t_2 = t1 * (-(v / u) / u);
	double tmp;
	if (t1 <= -125000000000.0) {
		tmp = t_1;
	} else if (t1 <= -5.8e-47) {
		tmp = t_2;
	} else if (t1 <= -8e-77) {
		tmp = t_1;
	} else if (t1 <= -8.2e-160) {
		tmp = t_2;
	} else if (t1 <= 3.4e-17) {
		tmp = v * ((t1 / -u) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / (t1 + u))
	t_2 = t1 * (-(v / u) / u)
	tmp = 0
	if t1 <= -125000000000.0:
		tmp = t_1
	elif t1 <= -5.8e-47:
		tmp = t_2
	elif t1 <= -8e-77:
		tmp = t_1
	elif t1 <= -8.2e-160:
		tmp = t_2
	elif t1 <= 3.4e-17:
		tmp = v * ((t1 / -u) / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / Float64(t1 + u)))
	t_2 = Float64(t1 * Float64(Float64(-Float64(v / u)) / u))
	tmp = 0.0
	if (t1 <= -125000000000.0)
		tmp = t_1;
	elseif (t1 <= -5.8e-47)
		tmp = t_2;
	elseif (t1 <= -8e-77)
		tmp = t_1;
	elseif (t1 <= -8.2e-160)
		tmp = t_2;
	elseif (t1 <= 3.4e-17)
		tmp = Float64(v * Float64(Float64(t1 / Float64(-u)) / u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(v / (t1 + u));
	t_2 = t1 * (-(v / u) / u);
	tmp = 0.0;
	if (t1 <= -125000000000.0)
		tmp = t_1;
	elseif (t1 <= -5.8e-47)
		tmp = t_2;
	elseif (t1 <= -8e-77)
		tmp = t_1;
	elseif (t1 <= -8.2e-160)
		tmp = t_2;
	elseif (t1 <= 3.4e-17)
		tmp = v * ((t1 / -u) / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$2 = N[(t1 * N[((-N[(v / u), $MachinePrecision]) / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -125000000000.0], t$95$1, If[LessEqual[t1, -5.8e-47], t$95$2, If[LessEqual[t1, -8e-77], t$95$1, If[LessEqual[t1, -8.2e-160], t$95$2, If[LessEqual[t1, 3.4e-17], N[(v * N[(N[(t1 / (-u)), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.25e11 or -5.8000000000000001e-47 < t1 < -7.9999999999999994e-77 or 3.3999999999999998e-17 < t1

   1. Initial program 68.8%

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

      1. associate-/r* 79.3%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   6. Simplified 83.8%

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

   if -1.25e11 < t1 < -5.8000000000000001e-47 or -7.9999999999999994e-77 < t1 < -8.20000000000000003e-160

   1. Initial program 74.6%

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

      1. associate-/r* 88.0%

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

      2. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

      1. div-inv 99.3%

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

      2. clear-num 99.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 29.5%

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

      5. sqr-neg 29.5%

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

      6. sqrt-unprod 29.5%

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

      7. add-sqr-sqrt 29.5%

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

      8. distribute-lft-neg-in 29.5%

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

      9. distribute-rgt-neg-in 29.5%

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

      10. add-sqr-sqrt 29.5%

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

      11. sqrt-unprod 29.5%

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

      12. sqr-neg 29.5%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in t1 around 0 63.1%

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

      1. mul-1-neg 63.1%

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

      2. associate-*r/ 71.0%

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

      3. unpow2 71.0%

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

      4. distribute-rgt-neg-in 71.0%

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

   8. Simplified 71.0%

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

   9. Taylor expanded in v around 0 71.0%

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

       1. unpow2 71.0%

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

       2. associate-/r* 84.4%

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

   11. Simplified 84.4%

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

   if -8.20000000000000003e-160 < t1 < 3.3999999999999998e-17

   1. Initial program 81.1%

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

      1. associate-*l/ 85.2%

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

      2. *-commutative 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in t1 around 0 79.0%

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

      1. associate-*r/ 79.0%

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

      2. neg-mul-1 79.0%

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

      3. unpow2 79.0%

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

   6. Simplified 79.0%

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

   7. Taylor expanded in t1 around 0 79.0%

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

      1. unpow2 79.0%

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

      2. associate-*r/ 79.0%

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

      3. times-frac 84.7%

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

      4. associate-*l/ 84.8%

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

      5. metadata-eval 84.8%

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

      6. times-frac 84.8%

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

      7. *-lft-identity 84.8%

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

      8. neg-mul-1 84.8%

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

   9. Simplified 84.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -125000000000:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -5.8 \cdot 10^{-47}:\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq -8 \cdot 10^{-77}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -8.2 \cdot 10^{-160}:\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \]

Alternative 4: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -16000000 \lor \neg \left(t1 \leq -6 \cdot 10^{-47}\right) \land \left(t1 \leq -8 \cdot 10^{-77} \lor \neg \left(t1 \leq 3.8 \cdot 10^{-17}\right)\right):\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -16000000.0)
         (and (not (<= t1 -6e-47)) (or (<= t1 -8e-77) (not (<= t1 3.8e-17)))))
   (- (/ v (+ t1 u)))
   (* t1 (/ (- v) (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -16000000.0) || (!(t1 <= -6e-47) && ((t1 <= -8e-77) || !(t1 <= 3.8e-17)))) {
		tmp = -(v / (t1 + u));
	} else {
		tmp = t1 * (-v / (u * 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 <= (-16000000.0d0)) .or. (.not. (t1 <= (-6d-47))) .and. (t1 <= (-8d-77)) .or. (.not. (t1 <= 3.8d-17))) then
        tmp = -(v / (t1 + u))
    else
        tmp = t1 * (-v / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -16000000.0) || (!(t1 <= -6e-47) && ((t1 <= -8e-77) || !(t1 <= 3.8e-17)))) {
		tmp = -(v / (t1 + u));
	} else {
		tmp = t1 * (-v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -16000000.0) or (not (t1 <= -6e-47) and ((t1 <= -8e-77) or not (t1 <= 3.8e-17))):
		tmp = -(v / (t1 + u))
	else:
		tmp = t1 * (-v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -16000000.0) || (!(t1 <= -6e-47) && ((t1 <= -8e-77) || !(t1 <= 3.8e-17))))
		tmp = Float64(-Float64(v / Float64(t1 + u)));
	else
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -16000000.0) || (~((t1 <= -6e-47)) && ((t1 <= -8e-77) || ~((t1 <= 3.8e-17)))))
		tmp = -(v / (t1 + u));
	else
		tmp = t1 * (-v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -16000000.0], And[N[Not[LessEqual[t1, -6e-47]], $MachinePrecision], Or[LessEqual[t1, -8e-77], N[Not[LessEqual[t1, 3.8e-17]], $MachinePrecision]]]], (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6e7 or -6.00000000000000033e-47 < t1 < -7.9999999999999994e-77 or 3.8000000000000001e-17 < t1

   1. Initial program 68.8%

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

      1. associate-/r* 79.3%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   6. Simplified 83.8%

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

   if -1.6e7 < t1 < -6.00000000000000033e-47 or -7.9999999999999994e-77 < t1 < 3.8000000000000001e-17

   1. Initial program 79.7%

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

      1. associate-/r* 87.6%

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

      2. associate-/l* 96.2%

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

   3. Simplified 96.2%

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

      1. div-inv 95.7%

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

      2. clear-num 95.8%

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

      3. add-sqr-sqrt 48.0%

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

      4. sqrt-unprod 55.4%

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

      5. sqr-neg 55.4%

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

      6. sqrt-unprod 17.8%

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

      7. add-sqr-sqrt 36.8%

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

      8. distribute-lft-neg-in 36.8%

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

      9. distribute-rgt-neg-in 36.8%

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

      10. add-sqr-sqrt 17.8%

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

      11. sqrt-unprod 55.4%

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

      12. sqr-neg 55.4%

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

      13. sqrt-unprod 48.0%

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

      14. add-sqr-sqrt 95.8%

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

   5. Applied egg-rr 95.8%

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

   6. Taylor expanded in t1 around 0 72.4%

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

      1. mul-1-neg 72.4%

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

      2. associate-*r/ 72.3%

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

      3. unpow2 72.3%

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

      4. distribute-rgt-neg-in 72.3%

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

   8. Simplified 72.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -16000000 \lor \neg \left(t1 \leq -6 \cdot 10^{-47}\right) \land \left(t1 \leq -8 \cdot 10^{-77} \lor \neg \left(t1 \leq 3.8 \cdot 10^{-17}\right)\right):\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 5: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -850000000000 \lor \neg \left(t1 \leq -6.2 \cdot 10^{-47} \lor \neg \left(t1 \leq -7.5 \cdot 10^{-77}\right) \land t1 \leq 1.35 \cdot 10^{-16}\right):\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -850000000000.0)
         (not
          (or (<= t1 -6.2e-47) (and (not (<= t1 -7.5e-77)) (<= t1 1.35e-16)))))
   (- (/ v (+ t1 u)))
   (* t1 (/ (- (/ v u)) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -850000000000.0) || !((t1 <= -6.2e-47) || (!(t1 <= -7.5e-77) && (t1 <= 1.35e-16)))) {
		tmp = -(v / (t1 + u));
	} else {
		tmp = t1 * (-(v / u) / 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 <= (-850000000000.0d0)) .or. (.not. (t1 <= (-6.2d-47)) .or. (.not. (t1 <= (-7.5d-77))) .and. (t1 <= 1.35d-16))) then
        tmp = -(v / (t1 + u))
    else
        tmp = t1 * (-(v / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -850000000000.0) || !((t1 <= -6.2e-47) || (!(t1 <= -7.5e-77) && (t1 <= 1.35e-16)))) {
		tmp = -(v / (t1 + u));
	} else {
		tmp = t1 * (-(v / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -850000000000.0) or not ((t1 <= -6.2e-47) or (not (t1 <= -7.5e-77) and (t1 <= 1.35e-16))):
		tmp = -(v / (t1 + u))
	else:
		tmp = t1 * (-(v / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -850000000000.0) || !((t1 <= -6.2e-47) || (!(t1 <= -7.5e-77) && (t1 <= 1.35e-16))))
		tmp = Float64(-Float64(v / Float64(t1 + u)));
	else
		tmp = Float64(t1 * Float64(Float64(-Float64(v / u)) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -850000000000.0) || ~(((t1 <= -6.2e-47) || (~((t1 <= -7.5e-77)) && (t1 <= 1.35e-16)))))
		tmp = -(v / (t1 + u));
	else
		tmp = t1 * (-(v / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -850000000000.0], N[Not[Or[LessEqual[t1, -6.2e-47], And[N[Not[LessEqual[t1, -7.5e-77]], $MachinePrecision], LessEqual[t1, 1.35e-16]]]], $MachinePrecision]], (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), N[(t1 * N[((-N[(v / u), $MachinePrecision]) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.5e11 or -6.1999999999999996e-47 < t1 < -7.5000000000000006e-77 or 1.35e-16 < t1

   1. Initial program 68.8%

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

      1. associate-/r* 79.3%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   6. Simplified 83.8%

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

   if -8.5e11 < t1 < -6.1999999999999996e-47 or -7.5000000000000006e-77 < t1 < 1.35e-16

   1. Initial program 79.7%

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

      1. associate-/r* 87.6%

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

      2. associate-/l* 96.2%

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

   3. Simplified 96.2%

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

      1. div-inv 95.7%

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

      2. clear-num 95.8%

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

      3. add-sqr-sqrt 48.0%

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

      4. sqrt-unprod 55.4%

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

      5. sqr-neg 55.4%

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

      6. sqrt-unprod 17.8%

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

      7. add-sqr-sqrt 36.8%

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

      8. distribute-lft-neg-in 36.8%

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

      9. distribute-rgt-neg-in 36.8%

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

      10. add-sqr-sqrt 17.8%

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

      11. sqrt-unprod 55.4%

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

      12. sqr-neg 55.4%

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

      13. sqrt-unprod 48.0%

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

      14. add-sqr-sqrt 95.8%

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

   5. Applied egg-rr 95.8%

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

   6. Taylor expanded in t1 around 0 72.4%

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

      1. mul-1-neg 72.4%

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

      2. associate-*r/ 72.3%

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

      3. unpow2 72.3%

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

      4. distribute-rgt-neg-in 72.3%

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

   8. Simplified 72.3%

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

   9. Taylor expanded in v around 0 72.3%

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

       1. unpow2 72.3%

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

       2. associate-/r* 77.9%

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

   11. Simplified 77.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -850000000000 \lor \neg \left(t1 \leq -6.2 \cdot 10^{-47} \lor \neg \left(t1 \leq -7.5 \cdot 10^{-77}\right) \land t1 \leq 1.35 \cdot 10^{-16}\right):\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \end{array} \]

Alternative 6: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -48000000 \lor \neg \left(t1 \leq -5 \cdot 10^{-47} \lor \neg \left(t1 \leq -8 \cdot 10^{-77}\right) \land t1 \leq 3.5 \cdot 10^{-16}\right):\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -48000000.0)
         (not (or (<= t1 -5e-47) (and (not (<= t1 -8e-77)) (<= t1 3.5e-16)))))
   (- (/ v (+ t1 u)))
   (* (/ v u) (/ t1 (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -48000000.0) || !((t1 <= -5e-47) || (!(t1 <= -8e-77) && (t1 <= 3.5e-16)))) {
		tmp = -(v / (t1 + u));
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-48000000.0d0)) .or. (.not. (t1 <= (-5d-47)) .or. (.not. (t1 <= (-8d-77))) .and. (t1 <= 3.5d-16))) then
        tmp = -(v / (t1 + u))
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -48000000.0) || !((t1 <= -5e-47) || (!(t1 <= -8e-77) && (t1 <= 3.5e-16)))) {
		tmp = -(v / (t1 + u));
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -48000000.0) or not ((t1 <= -5e-47) or (not (t1 <= -8e-77) and (t1 <= 3.5e-16))):
		tmp = -(v / (t1 + u))
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -48000000.0) || !((t1 <= -5e-47) || (!(t1 <= -8e-77) && (t1 <= 3.5e-16))))
		tmp = Float64(-Float64(v / Float64(t1 + u)));
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -48000000.0) || ~(((t1 <= -5e-47) || (~((t1 <= -8e-77)) && (t1 <= 3.5e-16)))))
		tmp = -(v / (t1 + u));
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -48000000.0], N[Not[Or[LessEqual[t1, -5e-47], And[N[Not[LessEqual[t1, -8e-77]], $MachinePrecision], LessEqual[t1, 3.5e-16]]]], $MachinePrecision]], (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.8e7 or -5.00000000000000011e-47 < t1 < -7.9999999999999994e-77 or 3.50000000000000017e-16 < t1

   1. Initial program 68.8%

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

      1. associate-/r* 79.3%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   6. Simplified 83.8%

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

   if -4.8e7 < t1 < -5.00000000000000011e-47 or -7.9999999999999994e-77 < t1 < 3.50000000000000017e-16

   1. Initial program 79.7%

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

      1. associate-/r* 87.6%

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

      2. associate-/l* 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in t1 around 0 93.6%

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

      1. +-commutative 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in t1 around 0 72.4%

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

      1. *-commutative 72.4%

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

      2. *-commutative 72.4%

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

      3. unpow2 72.4%

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

      4. times-frac 83.5%

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

      5. associate-*l* 83.5%

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

      6. metadata-eval 83.5%

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

      7. times-frac 83.5%

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

      8. *-rgt-identity 83.5%

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

      9. *-commutative 83.5%

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

      10. neg-mul-1 83.5%

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

   9. Simplified 83.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -48000000 \lor \neg \left(t1 \leq -5 \cdot 10^{-47} \lor \neg \left(t1 \leq -8 \cdot 10^{-77}\right) \land t1 \leq 3.5 \cdot 10^{-16}\right):\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]

Alternative 7: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{+33} \lor \neg \left(t1 \leq -1.1 \cdot 10^{-46} \lor \neg \left(t1 \leq -7.8 \cdot 10^{-77}\right) \land t1 \leq 1.85 \cdot 10^{-16}\right):\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-v\right) \cdot \frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4e+33)
         (not
          (or (<= t1 -1.1e-46) (and (not (<= t1 -7.8e-77)) (<= t1 1.85e-16)))))
   (- (/ v (+ t1 u)))
   (/ (* (- v) (/ t1 u)) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4e+33) || !((t1 <= -1.1e-46) || (!(t1 <= -7.8e-77) && (t1 <= 1.85e-16)))) {
		tmp = -(v / (t1 + u));
	} else {
		tmp = (-v * (t1 / u)) / 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 <= (-4d+33)) .or. (.not. (t1 <= (-1.1d-46)) .or. (.not. (t1 <= (-7.8d-77))) .and. (t1 <= 1.85d-16))) then
        tmp = -(v / (t1 + u))
    else
        tmp = (-v * (t1 / u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4e+33) || !((t1 <= -1.1e-46) || (!(t1 <= -7.8e-77) && (t1 <= 1.85e-16)))) {
		tmp = -(v / (t1 + u));
	} else {
		tmp = (-v * (t1 / u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4e+33) or not ((t1 <= -1.1e-46) or (not (t1 <= -7.8e-77) and (t1 <= 1.85e-16))):
		tmp = -(v / (t1 + u))
	else:
		tmp = (-v * (t1 / u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4e+33) || !((t1 <= -1.1e-46) || (!(t1 <= -7.8e-77) && (t1 <= 1.85e-16))))
		tmp = Float64(-Float64(v / Float64(t1 + u)));
	else
		tmp = Float64(Float64(Float64(-v) * Float64(t1 / u)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4e+33) || ~(((t1 <= -1.1e-46) || (~((t1 <= -7.8e-77)) && (t1 <= 1.85e-16)))))
		tmp = -(v / (t1 + u));
	else
		tmp = (-v * (t1 / u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4e+33], N[Not[Or[LessEqual[t1, -1.1e-46], And[N[Not[LessEqual[t1, -7.8e-77]], $MachinePrecision], LessEqual[t1, 1.85e-16]]]], $MachinePrecision]], (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), N[(N[((-v) * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.9999999999999998e33 or -1.1e-46 < t1 < -7.79999999999999958e-77 or 1.85e-16 < t1

   1. Initial program 68.6%

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

      1. associate-/r* 78.7%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 84.8%

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

      1. neg-mul-1 84.8%

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

   6. Simplified 84.8%

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

   if -3.9999999999999998e33 < t1 < -1.1e-46 or -7.79999999999999958e-77 < t1 < 1.85e-16

   1. Initial program 79.7%

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

      1. associate-/r* 88.0%

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

      2. associate-/l* 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in t1 around 0 93.8%

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

      1. +-commutative 93.8%

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

   6. Simplified 93.8%

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

   7. Taylor expanded in t1 around 0 71.0%

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

      1. *-commutative 71.0%

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

      2. *-commutative 71.0%

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

      3. unpow2 71.0%

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

      4. times-frac 82.5%

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

      5. associate-*l* 82.5%

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

      6. metadata-eval 82.5%

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

      7. times-frac 82.5%

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

      8. *-rgt-identity 82.5%

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

      9. *-commutative 82.5%

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

      10. neg-mul-1 82.5%

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

   9. Simplified 82.5%

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

       1. associate-*l/ 83.4%

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

       2. frac-2neg 83.4%

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

       3. add-sqr-sqrt 39.8%

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

       4. sqrt-unprod 54.2%

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

       5. sqr-neg 54.2%

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

       6. sqrt-prod 18.3%

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

       7. add-sqr-sqrt 36.5%

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

       8. add-sqr-sqrt 18.2%

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

       9. sqrt-unprod 57.3%

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

       10. sqr-neg 57.3%

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

       11. sqrt-prod 43.4%

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

       12. add-sqr-sqrt 83.4%

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

   11. Applied egg-rr 83.4%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{+33} \lor \neg \left(t1 \leq -1.1 \cdot 10^{-46} \lor \neg \left(t1 \leq -7.8 \cdot 10^{-77}\right) \land t1 \leq 1.85 \cdot 10^{-16}\right):\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-v\right) \cdot \frac{t1}{u}}{u}\\ \end{array} \]

Alternative 8: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -18000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.05 \cdot 10^{-46}:\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq -7.8 \cdot 10^{-77} \lor \neg \left(t1 \leq 4.6 \cdot 10^{-16}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (+ t1 u)))))
   (if (<= t1 -18000000.0)
     t_1
     (if (<= t1 -1.05e-46)
       (* t1 (/ (- (/ v u)) u))
       (if (or (<= t1 -7.8e-77) (not (<= t1 4.6e-16)))
         t_1
         (* v (/ (- t1) (* u u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / (t1 + u));
	double tmp;
	if (t1 <= -18000000.0) {
		tmp = t_1;
	} else if (t1 <= -1.05e-46) {
		tmp = t1 * (-(v / u) / u);
	} else if ((t1 <= -7.8e-77) || !(t1 <= 4.6e-16)) {
		tmp = t_1;
	} else {
		tmp = v * (-t1 / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -(v / (t1 + u))
    if (t1 <= (-18000000.0d0)) then
        tmp = t_1
    else if (t1 <= (-1.05d-46)) then
        tmp = t1 * (-(v / u) / u)
    else if ((t1 <= (-7.8d-77)) .or. (.not. (t1 <= 4.6d-16))) then
        tmp = t_1
    else
        tmp = v * (-t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -(v / (t1 + u));
	double tmp;
	if (t1 <= -18000000.0) {
		tmp = t_1;
	} else if (t1 <= -1.05e-46) {
		tmp = t1 * (-(v / u) / u);
	} else if ((t1 <= -7.8e-77) || !(t1 <= 4.6e-16)) {
		tmp = t_1;
	} else {
		tmp = v * (-t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / (t1 + u))
	tmp = 0
	if t1 <= -18000000.0:
		tmp = t_1
	elif t1 <= -1.05e-46:
		tmp = t1 * (-(v / u) / u)
	elif (t1 <= -7.8e-77) or not (t1 <= 4.6e-16):
		tmp = t_1
	else:
		tmp = v * (-t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -18000000.0)
		tmp = t_1;
	elseif (t1 <= -1.05e-46)
		tmp = Float64(t1 * Float64(Float64(-Float64(v / u)) / u));
	elseif ((t1 <= -7.8e-77) || !(t1 <= 4.6e-16))
		tmp = t_1;
	else
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(v / (t1 + u));
	tmp = 0.0;
	if (t1 <= -18000000.0)
		tmp = t_1;
	elseif (t1 <= -1.05e-46)
		tmp = t1 * (-(v / u) / u);
	elseif ((t1 <= -7.8e-77) || ~((t1 <= 4.6e-16)))
		tmp = t_1;
	else
		tmp = v * (-t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -18000000.0], t$95$1, If[LessEqual[t1, -1.05e-46], N[(t1 * N[((-N[(v / u), $MachinePrecision]) / u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, -7.8e-77], N[Not[LessEqual[t1, 4.6e-16]], $MachinePrecision]], t$95$1, N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.8e7 or -1.04999999999999994e-46 < t1 < -7.79999999999999958e-77 or 4.5999999999999998e-16 < t1

   1. Initial program 68.8%

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

      1. associate-/r* 79.3%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   6. Simplified 83.8%

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

   if -1.8e7 < t1 < -1.04999999999999994e-46

   1. Initial program 85.8%

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

      1. associate-/r* 99.7%

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

      2. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

      1. div-inv 99.5%

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

      2. clear-num 99.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 27.5%

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

      5. sqr-neg 27.5%

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

      6. sqrt-unprod 27.5%

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

      7. add-sqr-sqrt 27.5%

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

      8. distribute-lft-neg-in 27.5%

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

      9. distribute-rgt-neg-in 27.5%

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

      10. add-sqr-sqrt 27.5%

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

      11. sqrt-unprod 27.5%

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

      12. sqr-neg 27.5%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in t1 around 0 76.1%

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

      1. mul-1-neg 76.1%

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

      2. associate-*r/ 76.0%

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

      3. unpow2 76.0%

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

      4. distribute-rgt-neg-in 76.0%

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

   8. Simplified 76.0%

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

   9. Taylor expanded in v around 0 76.0%

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

       1. unpow2 76.0%

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

       2. associate-/r* 89.5%

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

   11. Simplified 89.5%

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

   if -7.79999999999999958e-77 < t1 < 4.5999999999999998e-16

   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. associate-*l/ 85.4%

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

      2. *-commutative 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in t1 around 0 77.5%

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

      1. associate-*r/ 77.5%

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

      2. neg-mul-1 77.5%

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

      3. unpow2 77.5%

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

   6. Simplified 77.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -18000000:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.05 \cdot 10^{-46}:\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq -7.8 \cdot 10^{-77} \lor \neg \left(t1 \leq 4.6 \cdot 10^{-16}\right):\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \]

Alternative 9: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\frac{u \cdot \frac{v}{t1} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -7 \cdot 10^{-47} \lor \neg \left(t1 \leq -8 \cdot 10^{-77}\right) \land t1 \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(-v\right) \cdot \frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4e+33)
   (/ (- (* u (/ v t1)) v) (+ t1 u))
   (if (or (<= t1 -7e-47) (and (not (<= t1 -8e-77)) (<= t1 3.5e-16)))
     (/ (* (- v) (/ t1 u)) u)
     (- (/ v (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4e+33) {
		tmp = ((u * (v / t1)) - v) / (t1 + u);
	} else if ((t1 <= -7e-47) || (!(t1 <= -8e-77) && (t1 <= 3.5e-16))) {
		tmp = (-v * (t1 / u)) / u;
	} else {
		tmp = -(v / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-4d+33)) then
        tmp = ((u * (v / t1)) - v) / (t1 + u)
    else if ((t1 <= (-7d-47)) .or. (.not. (t1 <= (-8d-77))) .and. (t1 <= 3.5d-16)) then
        tmp = (-v * (t1 / u)) / u
    else
        tmp = -(v / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4e+33) {
		tmp = ((u * (v / t1)) - v) / (t1 + u);
	} else if ((t1 <= -7e-47) || (!(t1 <= -8e-77) && (t1 <= 3.5e-16))) {
		tmp = (-v * (t1 / u)) / u;
	} else {
		tmp = -(v / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4e+33:
		tmp = ((u * (v / t1)) - v) / (t1 + u)
	elif (t1 <= -7e-47) or (not (t1 <= -8e-77) and (t1 <= 3.5e-16)):
		tmp = (-v * (t1 / u)) / u
	else:
		tmp = -(v / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4e+33)
		tmp = Float64(Float64(Float64(u * Float64(v / t1)) - v) / Float64(t1 + u));
	elseif ((t1 <= -7e-47) || (!(t1 <= -8e-77) && (t1 <= 3.5e-16)))
		tmp = Float64(Float64(Float64(-v) * Float64(t1 / 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 <= -4e+33)
		tmp = ((u * (v / t1)) - v) / (t1 + u);
	elseif ((t1 <= -7e-47) || (~((t1 <= -8e-77)) && (t1 <= 3.5e-16)))
		tmp = (-v * (t1 / u)) / u;
	else
		tmp = -(v / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4e+33], N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, -7e-47], And[N[Not[LessEqual[t1, -8e-77]], $MachinePrecision], LessEqual[t1, 3.5e-16]]], N[(N[((-v) * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.9999999999999998e33

   1. Initial program 65.0%

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

      1. associate-/r* 76.2%

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

      2. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around 0 91.5%

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

      1. +-commutative 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in t1 around inf 90.5%

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

      1. neg-mul-1 90.5%

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

      2. associate-*r/ 92.4%

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

      3. +-commutative 92.4%

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

      4. sub-neg 92.4%

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

   9. Simplified 92.4%

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

   if -3.9999999999999998e33 < t1 < -6.9999999999999996e-47 or -7.9999999999999994e-77 < t1 < 3.50000000000000017e-16

   1. Initial program 79.7%

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

      1. associate-/r* 88.0%

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

      2. associate-/l* 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in t1 around 0 93.8%

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

      1. +-commutative 93.8%

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

   6. Simplified 93.8%

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

   7. Taylor expanded in t1 around 0 71.0%

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

      1. *-commutative 71.0%

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

      2. *-commutative 71.0%

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

      3. unpow2 71.0%

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

      4. times-frac 82.5%

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

      5. associate-*l* 82.5%

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

      6. metadata-eval 82.5%

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

      7. times-frac 82.5%

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

      8. *-rgt-identity 82.5%

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

      9. *-commutative 82.5%

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

      10. neg-mul-1 82.5%

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

   9. Simplified 82.5%

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

       1. associate-*l/ 83.4%

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

       2. frac-2neg 83.4%

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

       3. add-sqr-sqrt 39.8%

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

       4. sqrt-unprod 54.2%

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

       5. sqr-neg 54.2%

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

       6. sqrt-prod 18.3%

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

       7. add-sqr-sqrt 36.5%

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

       8. add-sqr-sqrt 18.2%

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

       9. sqrt-unprod 57.3%

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

       10. sqr-neg 57.3%

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

       11. sqrt-prod 43.4%

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

       12. add-sqr-sqrt 83.4%

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

   11. Applied egg-rr 83.4%

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

   if -6.9999999999999996e-47 < t1 < -7.9999999999999994e-77 or 3.50000000000000017e-16 < t1

   1. Initial program 70.9%

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

      1. associate-/r* 80.2%

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

      2. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t1 around inf 80.0%

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

      1. neg-mul-1 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\frac{u \cdot \frac{v}{t1} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -7 \cdot 10^{-47} \lor \neg \left(t1 \leq -8 \cdot 10^{-77}\right) \land t1 \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(-v\right) \cdot \frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \]

Alternative 10: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{t1}{\frac{t1 - u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 42:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.9e-115)
   (/ t1 (/ (- t1 u) (/ v u)))
   (if (<= u 42.0) (- (/ v t1)) (/ (/ t1 (+ t1 u)) (/ (- t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.9e-115) {
		tmp = t1 / ((t1 - u) / (v / u));
	} else if (u <= 42.0) {
		tmp = -(v / t1);
	} else {
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.9d-115)) then
        tmp = t1 / ((t1 - u) / (v / u))
    else if (u <= 42.0d0) then
        tmp = -(v / t1)
    else
        tmp = (t1 / (t1 + u)) / ((t1 - u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.9e-115) {
		tmp = t1 / ((t1 - u) / (v / u));
	} else if (u <= 42.0) {
		tmp = -(v / t1);
	} else {
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.9e-115:
		tmp = t1 / ((t1 - u) / (v / u))
	elif u <= 42.0:
		tmp = -(v / t1)
	else:
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.9e-115)
		tmp = Float64(t1 / Float64(Float64(t1 - u) / Float64(v / u)));
	elseif (u <= 42.0)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) / Float64(Float64(t1 - u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.9e-115)
		tmp = t1 / ((t1 - u) / (v / u));
	elseif (u <= 42.0)
		tmp = -(v / t1);
	else
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.9e-115], N[(t1 / N[(N[(t1 - u), $MachinePrecision] / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 42.0], (-N[(v / t1), $MachinePrecision]), N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.8999999999999998e-115

   1. Initial program 70.4%

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

      1. associate-*l/ 77.7%

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

      2. *-commutative 77.7%

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

   3. Simplified 77.7%

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

      1. associate-/r* 91.8%

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

      2. associate-*r/ 97.0%

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

      3. *-commutative 97.0%

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

      4. associate-/r/ 95.9%

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

      5. frac-2neg 95.9%

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

      6. distribute-frac-neg 95.9%

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

      7. remove-double-neg 95.9%

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

      8. div-inv 95.8%

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

      9. clear-num 95.9%

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

      10. distribute-neg-in 95.9%

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

      11. add-sqr-sqrt 44.2%

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

      12. sqrt-unprod 81.8%

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

      13. sqr-neg 81.8%

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

      14. sqrt-unprod 43.8%

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

      15. add-sqr-sqrt 75.5%

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

      16. sub-neg 75.5%

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

   5. Applied egg-rr 75.5%

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

      1. associate-/l* 77.4%

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

   7. Simplified 77.4%

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

   8. Taylor expanded in t1 around 0 77.4%

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

   if -2.8999999999999998e-115 < u < 42

   1. Initial program 72.0%

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

      1. associate-*l/ 77.8%

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

      2. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in t1 around inf 81.2%

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

      1. associate-*r/ 81.2%

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

      2. neg-mul-1 81.2%

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

   6. Simplified 81.2%

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

   if 42 < u

   1. Initial program 81.9%

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

      1. associate-*l/ 80.4%

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

      2. *-commutative 80.4%

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

   3. Simplified 80.4%

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

      1. associate-/r* 86.7%

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

      2. associate-*r/ 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-/r/ 99.0%

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

      5. div-inv 99.0%

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

      6. clear-num 98.9%

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

      7. associate-/r/ 98.9%

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

      8. clear-num 99.8%

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

      9. *-commutative 99.8%

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

      10. clear-num 98.9%

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

      11. div-inv 99.0%

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

      12. frac-2neg 99.0%

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

      13. remove-double-neg 99.0%

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

      14. associate-*l/ 98.3%

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

   5. Applied egg-rr 90.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{t1}{\frac{t1 - u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 42:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1 - u}{v}}\\ \end{array} \]

Alternative 11: 68.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+87} \lor \neg \left(u \leq 5.1 \cdot 10^{+101}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.55e+87) (not (<= u 5.1e+101)))
   (* v (/ t1 (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.55e+87) || !(u <= 5.1e+101)) {
		tmp = v * (t1 / (u * 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 <= (-1.55d+87)) .or. (.not. (u <= 5.1d+101))) then
        tmp = v * (t1 / (u * 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 <= -1.55e+87) || !(u <= 5.1e+101)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.55e+87) or not (u <= 5.1e+101):
		tmp = v * (t1 / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.55e+87) || !(u <= 5.1e+101))
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.55e+87) || ~((u <= 5.1e+101)))
		tmp = v * (t1 / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.55e+87], N[Not[LessEqual[u, 5.1e+101]], $MachinePrecision]], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.55e87 or 5.09999999999999995e101 < u

   1. Initial program 74.0%

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

   2. Taylor expanded in t1 around 0 73.0%

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

      1. unpow2 73.0%

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

   4. Simplified 73.0%

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

      1. associate-/l* 74.9%

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

      2. associate-/r/ 72.1%

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

      3. add-sqr-sqrt 33.4%

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

      4. sqrt-unprod 48.6%

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

      5. sqr-neg 48.6%

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

      6. sqrt-unprod 34.4%

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

      7. add-sqr-sqrt 66.0%

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

   6. Applied egg-rr 66.0%

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

   if -1.55e87 < u < 5.09999999999999995e101

   1. Initial program 73.6%

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

      1. associate-*l/ 81.2%

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

      2. *-commutative 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in t1 around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   6. Simplified 68.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+87} \lor \neg \left(u \leq 5.1 \cdot 10^{+101}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12: 68.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+85}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.8e+85)
   (* v (/ t1 (* u u)))
   (if (<= u 1.3e+106) (/ (- v) t1) (/ v (/ (* u u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.8e+85) {
		tmp = v * (t1 / (u * u));
	} else if (u <= 1.3e+106) {
		tmp = -v / t1;
	} else {
		tmp = v / ((u * u) / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.8d+85)) then
        tmp = v * (t1 / (u * u))
    else if (u <= 1.3d+106) then
        tmp = -v / t1
    else
        tmp = v / ((u * u) / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.8e+85) {
		tmp = v * (t1 / (u * u));
	} else if (u <= 1.3e+106) {
		tmp = -v / t1;
	} else {
		tmp = v / ((u * u) / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.8e+85:
		tmp = v * (t1 / (u * u))
	elif u <= 1.3e+106:
		tmp = -v / t1
	else:
		tmp = v / ((u * u) / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.8e+85)
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	elseif (u <= 1.3e+106)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / Float64(Float64(u * u) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.8e+85)
		tmp = v * (t1 / (u * u));
	elseif (u <= 1.3e+106)
		tmp = -v / t1;
	else
		tmp = v / ((u * u) / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.8e+85], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.3e+106], N[((-v) / t1), $MachinePrecision], N[(v / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.79999999999999993e85

   1. Initial program 70.4%

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

   2. Taylor expanded in t1 around 0 70.4%

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

      1. unpow2 70.4%

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

   4. Simplified 70.4%

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

      1. associate-/l* 72.6%

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

      2. associate-/r/ 72.6%

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

      3. add-sqr-sqrt 27.5%

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

      4. sqrt-unprod 44.4%

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

      5. sqr-neg 44.4%

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

      6. sqrt-unprod 37.3%

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

      7. add-sqr-sqrt 64.7%

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

   6. Applied egg-rr 64.7%

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

   if -4.79999999999999993e85 < u < 1.3000000000000001e106

   1. Initial program 73.6%

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

      1. associate-*l/ 81.2%

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

      2. *-commutative 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in t1 around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   6. Simplified 68.2%

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

   if 1.3000000000000001e106 < u

   1. Initial program 78.4%

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

   2. Taylor expanded in t1 around 0 76.3%

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

      1. unpow2 76.3%

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

   4. Simplified 76.3%

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

      1. associate-/l* 77.7%

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

      2. associate-/r/ 71.5%

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

      3. add-sqr-sqrt 40.8%

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

      4. sqrt-unprod 53.7%

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

      5. sqr-neg 53.7%

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

      6. sqrt-unprod 30.7%

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

      7. add-sqr-sqrt 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. *-commutative 67.5%

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

      2. clear-num 67.6%

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

      3. un-div-inv 67.6%

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

   8. Applied egg-rr 67.6%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+85}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \end{array} \]

Alternative 13: 24.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.16 \cdot 10^{+55}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{+88}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.16e+55) (/ v t1) (if (<= t1 1.9e+88) (/ v u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.16e+55) {
		tmp = v / t1;
	} else if (t1 <= 1.9e+88) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.16d+55)) then
        tmp = v / t1
    else if (t1 <= 1.9d+88) then
        tmp = v / u
    else
        tmp = v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.16e+55) {
		tmp = v / t1;
	} else if (t1 <= 1.9e+88) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.16e+55:
		tmp = v / t1
	elif t1 <= 1.9e+88:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.16e+55)
		tmp = Float64(v / t1);
	elseif (t1 <= 1.9e+88)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.16e+55)
		tmp = v / t1;
	elseif (t1 <= 1.9e+88)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.16e+55], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 1.9e+88], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.1599999999999999e55 or 1.8999999999999998e88 < t1

   1. Initial program 63.0%

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

      1. associate-/r* 72.8%

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

      2. associate-/l* 98.8%

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

   3. Simplified 98.8%

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

      1. div-inv 98.7%

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

      2. clear-num 100.0%

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

      3. add-sqr-sqrt 47.1%

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

      4. sqrt-unprod 22.4%

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

      5. sqr-neg 22.4%

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

      6. sqrt-unprod 24.6%

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

      7. add-sqr-sqrt 44.5%

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

      8. distribute-lft-neg-in 44.5%

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

      9. distribute-rgt-neg-in 44.5%

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

      10. add-sqr-sqrt 24.6%

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

      11. sqrt-unprod 22.4%

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

      12. sqr-neg 22.4%

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

      13. sqrt-unprod 47.1%

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

      14. add-sqr-sqrt 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in t1 around inf 91.0%

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

      1. expm1-log1p-u 75.4%

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

      2. expm1-udef 60.3%

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

      3. add-sqr-sqrt 50.8%

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

      4. sqrt-unprod 51.9%

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

      5. sqr-neg 51.9%

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

      6. sqrt-unprod 38.4%

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

      7. add-sqr-sqrt 41.4%

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

      8. associate-*r/ 41.2%

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

   8. Applied egg-rr 41.2%

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

      1. expm1-def 40.8%

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

      2. expm1-log1p 41.1%

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

      3. associate-/l* 41.6%

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

      4. associate-/r/ 41.5%

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

      5. *-inverses 41.5%

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

      6. *-lft-identity 41.5%

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

   10. Simplified 41.5%

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

   11. Taylor expanded in t1 around inf 40.4%

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

   if -1.1599999999999999e55 < t1 < 1.8999999999999998e88

   1. Initial program 79.9%

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

      1. associate-/r* 88.8%

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

      2. associate-/l* 97.2%

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

   3. Simplified 97.2%

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

      1. div-inv 96.8%

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

      2. clear-num 97.0%

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

      3. add-sqr-sqrt 52.1%

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

      4. sqrt-unprod 59.5%

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

      5. sqr-neg 59.5%

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

      6. sqrt-unprod 14.6%

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

      7. add-sqr-sqrt 29.8%

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

      8. distribute-lft-neg-in 29.8%

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

      9. distribute-rgt-neg-in 29.8%

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

      10. add-sqr-sqrt 14.6%

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

      11. sqrt-unprod 59.5%

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

      12. sqr-neg 59.5%

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

      13. sqrt-unprod 52.1%

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

      14. add-sqr-sqrt 97.0%

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

   5. Applied egg-rr 97.0%

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

   6. Taylor expanded in t1 around inf 41.6%

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

      1. expm1-log1p-u 34.9%

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

      2. expm1-udef 28.8%

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

      3. add-sqr-sqrt 13.6%

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

      4. sqrt-unprod 21.9%

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

      5. sqr-neg 21.9%

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

      6. sqrt-unprod 12.1%

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

      7. add-sqr-sqrt 19.8%

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

      8. associate-*r/ 19.9%

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

   8. Applied egg-rr 19.9%

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

      1. expm1-def 15.8%

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

      2. expm1-log1p 17.3%

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

      3. associate-/l* 12.9%

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

      4. associate-/r/ 13.0%

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

      5. *-inverses 13.0%

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

      6. *-lft-identity 13.0%

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

   10. Simplified 13.0%

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

   11. Taylor expanded in t1 around 0 15.2%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.16 \cdot 10^{+55}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{+88}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 14: 56.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 1.1 \cdot 10^{+109}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 1.1e+109) (- (/ v t1)) (/ v (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.1e+109) {
		tmp = -(v / t1);
	} 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 <= 1.1d+109) then
        tmp = -(v / t1)
    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 <= 1.1e+109) {
		tmp = -(v / t1);
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 1.1e+109:
		tmp = -(v / t1)
	else:
		tmp = v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 1.1e+109)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(v / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 1.1e+109)
		tmp = -(v / t1);
	else
		tmp = v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 1.1e+109], (-N[(v / t1), $MachinePrecision]), N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 1.1e109

   1. Initial program 72.9%

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

      1. associate-*l/ 79.3%

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

      2. *-commutative 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in t1 around inf 59.4%

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

      1. associate-*r/ 59.4%

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

      2. neg-mul-1 59.4%

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

   6. Simplified 59.4%

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

   if 1.1e109 < u

   1. Initial program 78.4%

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

      1. associate-/r* 87.7%

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

      2. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

      1. div-inv 98.6%

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

      2. clear-num 99.8%

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

      3. add-sqr-sqrt 39.9%

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

      4. sqrt-unprod 55.1%

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

      5. sqr-neg 55.1%

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

      6. sqrt-unprod 37.4%

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

      7. add-sqr-sqrt 68.0%

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

      8. distribute-lft-neg-in 68.0%

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

      9. distribute-rgt-neg-in 68.0%

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

      10. add-sqr-sqrt 37.4%

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

      11. sqrt-unprod 55.1%

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

      12. sqr-neg 55.1%

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

      13. sqrt-unprod 39.9%

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

      14. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in t1 around inf 48.7%

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

      1. expm1-log1p-u 46.2%

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

      2. expm1-udef 49.9%

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

      3. add-sqr-sqrt 31.6%

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

      4. sqrt-unprod 47.0%

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

      5. sqr-neg 47.0%

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

      6. sqrt-unprod 38.1%

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

      7. add-sqr-sqrt 50.3%

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

      8. associate-*r/ 50.1%

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

   8. Applied egg-rr 50.1%

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

      1. expm1-def 48.2%

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

      2. expm1-log1p 49.0%

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

      3. associate-/l* 46.8%

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

      4. associate-/r/ 46.9%

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

      5. *-inverses 46.9%

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

      6. *-lft-identity 46.9%

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

   10. Simplified 46.9%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 1.1 \cdot 10^{+109}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]

Alternative 15: 56.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 9.5 \cdot 10^{+103}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= u 9.5e+103) (- (/ v t1)) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 9.5e+103) {
		tmp = -(v / t1);
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 9.5d+103) then
        tmp = -(v / t1)
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 9.5e+103) {
		tmp = -(v / t1);
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 9.5e+103:
		tmp = -(v / t1)
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 9.5e+103)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 9.5e+103)
		tmp = -(v / t1);
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 9.5e+103], (-N[(v / t1), $MachinePrecision]), N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 9.49999999999999922e103

   1. Initial program 72.9%

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

      1. associate-*l/ 79.3%

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

      2. *-commutative 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in t1 around inf 59.4%

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

      1. associate-*r/ 59.4%

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

      2. neg-mul-1 59.4%

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

   6. Simplified 59.4%

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

   if 9.49999999999999922e103 < u

   1. Initial program 78.4%

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

      1. associate-/r* 87.7%

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

      2. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

      1. div-inv 98.6%

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

      2. clear-num 99.8%

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

      3. add-sqr-sqrt 39.9%

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

      4. sqrt-unprod 55.1%

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

      5. sqr-neg 55.1%

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

      6. sqrt-unprod 37.4%

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

      7. add-sqr-sqrt 68.0%

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

      8. distribute-lft-neg-in 68.0%

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

      9. distribute-rgt-neg-in 68.0%

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

      10. add-sqr-sqrt 37.4%

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

      11. sqrt-unprod 55.1%

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

      12. sqr-neg 55.1%

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

      13. sqrt-unprod 39.9%

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

      14. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in t1 around inf 48.7%

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

      1. expm1-log1p-u 46.2%

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

      2. expm1-udef 49.9%

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

      3. add-sqr-sqrt 31.6%

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

      4. sqrt-unprod 47.0%

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

      5. sqr-neg 47.0%

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

      6. sqrt-unprod 38.1%

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

      7. add-sqr-sqrt 50.3%

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

      8. associate-*r/ 50.1%

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

   8. Applied egg-rr 50.1%

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

      1. expm1-def 48.2%

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

      2. expm1-log1p 49.0%

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

      3. associate-/l* 46.8%

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

      4. associate-/r/ 46.9%

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

      5. *-inverses 46.9%

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

      6. *-lft-identity 46.9%

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

   10. Simplified 46.9%

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

   11. Taylor expanded in t1 around 0 44.1%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 9.5 \cdot 10^{+103}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 16: 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 73.7%

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

   1. associate-/r* 83.0%

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

   2. associate-/l* 97.8%

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

3. Simplified 97.8%

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

4. Taylor expanded in t1 around inf 59.6%

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

   1. neg-mul-1 59.6%

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

6. Simplified 59.6%

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

7. Final simplification 59.6%

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

Alternative 17: 14.3% 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 73.7%

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

   1. associate-/r* 83.0%

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

   2. associate-/l* 97.8%

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

3. Simplified 97.8%

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

   1. div-inv 97.5%

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

   2. clear-num 98.1%

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

   3. add-sqr-sqrt 50.3%

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

   4. sqrt-unprod 46.0%

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

   5. sqr-neg 46.0%

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

   6. sqrt-unprod 18.2%

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

   7. add-sqr-sqrt 35.1%

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

   8. distribute-lft-neg-in 35.1%

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

   9. distribute-rgt-neg-in 35.1%

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

   10. add-sqr-sqrt 18.2%

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

   11. sqrt-unprod 46.0%

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

   12. sqr-neg 46.0%

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

   13. sqrt-unprod 50.3%

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

   14. add-sqr-sqrt 98.1%

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

5. Applied egg-rr 98.1%

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

6. Taylor expanded in t1 around inf 59.5%

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

   1. expm1-log1p-u 49.6%

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

   2. expm1-udef 40.3%

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

   3. add-sqr-sqrt 27.1%

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

   4. sqrt-unprod 32.8%

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

   5. sqr-neg 32.8%

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

   6. sqrt-unprod 21.6%

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

   7. add-sqr-sqrt 27.7%

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

   8. associate-*r/ 27.6%

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

8. Applied egg-rr 27.6%

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

   1. expm1-def 24.9%

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

   2. expm1-log1p 25.9%

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

   3. associate-/l* 23.3%

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

   4. associate-/r/ 23.4%

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

   5. *-inverses 23.4%

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

   6. *-lft-identity 23.4%

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

10. Simplified 23.4%

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

11. Taylor expanded in t1 around inf 16.3%

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

12. Final simplification 16.3%

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

Reproduce

herbie shell --seed 2023279 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))