Rosa's DopplerBench

Percentage Accurate: 72.3% → 97.8%

Time: 9.0s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. times-frac 97.2%

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

3. Simplified 97.2%

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

4. Final simplification 97.2%

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

Alternative 2: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1}\\ \mathbf{if}\;u \leq -1600000000000:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq -9 \cdot 10^{-106}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 8 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.02 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{v}{\frac{t1 - u}{t1}}}{t1 + u}\\ \mathbf{elif}\;u \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{\left(-t1\right) - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v t1))))
   (if (<= u -1600000000000.0)
     (* (/ v (+ t1 u)) (/ (- t1) u))
     (if (<= u -9e-106)
       (/ v (- (* u -2.0) t1))
       (if (<= u -2.7e-151)
         (/ v (* (/ u t1) (- t1 u)))
         (if (<= u 8e-130)
           t_1
           (if (<= u 1.02e-83)
             (/ (/ v (/ (- t1 u) t1)) (+ t1 u))
             (if (<= u 2.7e+23) t_1 (/ t1 (* u (/ (- (- t1) u) v)))))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / t1);
	double tmp;
	if (u <= -1600000000000.0) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else if (u <= -9e-106) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -2.7e-151) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 8e-130) {
		tmp = t_1;
	} else if (u <= 1.02e-83) {
		tmp = (v / ((t1 - u) / t1)) / (t1 + u);
	} else if (u <= 2.7e+23) {
		tmp = t_1;
	} else {
		tmp = 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)
    if (u <= (-1600000000000.0d0)) then
        tmp = (v / (t1 + u)) * (-t1 / u)
    else if (u <= (-9d-106)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if (u <= (-2.7d-151)) then
        tmp = v / ((u / t1) * (t1 - u))
    else if (u <= 8d-130) then
        tmp = t_1
    else if (u <= 1.02d-83) then
        tmp = (v / ((t1 - u) / t1)) / (t1 + u)
    else if (u <= 2.7d+23) then
        tmp = t_1
    else
        tmp = 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);
	double tmp;
	if (u <= -1600000000000.0) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else if (u <= -9e-106) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -2.7e-151) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 8e-130) {
		tmp = t_1;
	} else if (u <= 1.02e-83) {
		tmp = (v / ((t1 - u) / t1)) / (t1 + u);
	} else if (u <= 2.7e+23) {
		tmp = t_1;
	} else {
		tmp = t1 / (u * ((-t1 - u) / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / t1)
	tmp = 0
	if u <= -1600000000000.0:
		tmp = (v / (t1 + u)) * (-t1 / u)
	elif u <= -9e-106:
		tmp = v / ((u * -2.0) - t1)
	elif u <= -2.7e-151:
		tmp = v / ((u / t1) * (t1 - u))
	elif u <= 8e-130:
		tmp = t_1
	elif u <= 1.02e-83:
		tmp = (v / ((t1 - u) / t1)) / (t1 + u)
	elif u <= 2.7e+23:
		tmp = t_1
	else:
		tmp = t1 / (u * ((-t1 - u) / v))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / t1))
	tmp = 0.0
	if (u <= -1600000000000.0)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	elseif (u <= -9e-106)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif (u <= -2.7e-151)
		tmp = Float64(v / Float64(Float64(u / t1) * Float64(t1 - u)));
	elseif (u <= 8e-130)
		tmp = t_1;
	elseif (u <= 1.02e-83)
		tmp = Float64(Float64(v / Float64(Float64(t1 - u) / t1)) / Float64(t1 + u));
	elseif (u <= 2.7e+23)
		tmp = t_1;
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(Float64(-t1) - u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(v / t1);
	tmp = 0.0;
	if (u <= -1600000000000.0)
		tmp = (v / (t1 + u)) * (-t1 / u);
	elseif (u <= -9e-106)
		tmp = v / ((u * -2.0) - t1);
	elseif (u <= -2.7e-151)
		tmp = v / ((u / t1) * (t1 - u));
	elseif (u <= 8e-130)
		tmp = t_1;
	elseif (u <= 1.02e-83)
		tmp = (v / ((t1 - u) / t1)) / (t1 + u);
	elseif (u <= 2.7e+23)
		tmp = t_1;
	else
		tmp = t1 / (u * ((-t1 - u) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / t1), $MachinePrecision])}, If[LessEqual[u, -1600000000000.0], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -9e-106], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -2.7e-151], N[(v / N[(N[(u / t1), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8e-130], t$95$1, If[LessEqual[u, 1.02e-83], N[(N[(v / N[(N[(t1 - u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.7e+23], t$95$1, N[(t1 / N[(u * N[(N[((-t1) - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if u < -1.6e12

   1. Initial program 76.3%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around 0 92.5%

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

      1. associate-*r/ 92.5%

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

      2. mul-1-neg 92.5%

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

   6. Simplified 92.5%

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

   if -1.6e12 < u < -8.99999999999999911e-106

   1. Initial program 82.0%

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

      1. associate-/r* 85.8%

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

      2. *-commutative 85.8%

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

      3. associate-/l* 99.7%

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

      4. associate-/l/ 99.7%

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

      5. +-commutative 99.7%

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

      6. remove-double-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. div-sub 99.7%

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

      9. sub-neg 99.7%

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

      10. *-inverses 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t1 around inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. unsub-neg 74.6%

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

      3. *-commutative 74.6%

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

   6. Simplified 74.6%

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

   if -8.99999999999999911e-106 < u < -2.70000000000000007e-151

   1. Initial program 61.5%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in t1 around 0 61.2%

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

      1. associate-*r/ 61.2%

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

      2. mul-1-neg 61.2%

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

   6. Simplified 61.2%

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

      1. associate-*r/ 61.3%

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

      2. frac-2neg 61.3%

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

      3. add-sqr-sqrt 10.5%

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

      4. sqrt-unprod 13.3%

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

      5. sqr-neg 13.3%

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

      6. sqrt-unprod 12.0%

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

      7. add-sqr-sqrt 12.7%

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

      8. distribute-lft-neg-out 12.7%

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

      9. distribute-frac-neg 12.7%

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

      10. clear-num 12.7%

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

      11. associate-*l/ 12.7%

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

      12. *-un-lft-identity 12.7%

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

      13. add-sqr-sqrt 0.7%

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

      14. sqrt-unprod 22.6%

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

      15. sqr-neg 22.6%

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

      16. sqrt-unprod 50.6%

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

      17. add-sqr-sqrt 61.0%

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

      18. distribute-neg-in 61.0%

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

   8. Applied egg-rr 61.3%

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

      1. associate-/l/ 80.2%

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

      2. *-commutative 80.2%

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

   10. Simplified 80.2%

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

   if -2.70000000000000007e-151 < u < 8.0000000000000007e-130 or 1.0199999999999999e-83 < u < 2.6999999999999999e23

   1. Initial program 62.8%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 84.6%

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

      1. associate-*r/ 84.6%

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

      2. neg-mul-1 84.6%

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

   6. Simplified 84.6%

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

   if 8.0000000000000007e-130 < u < 1.0199999999999999e-83

   1. Initial program 89.9%

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

      1. times-frac 90.1%

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

   3. Simplified 90.1%

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

      1. associate-*r/ 96.4%

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

      2. clear-num 96.1%

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

      3. associate-*l/ 96.1%

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

      4. *-un-lft-identity 96.1%

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

      5. frac-2neg 96.1%

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

      6. distribute-neg-in 96.1%

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

      7. add-sqr-sqrt 49.4%

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

      8. sqrt-unprod 94.3%

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

      9. sqr-neg 94.3%

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

      10. sqrt-unprod 44.6%

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

      11. add-sqr-sqrt 74.5%

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

      12. sub-neg 74.5%

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

      13. remove-double-neg 74.5%

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

   5. Applied egg-rr 74.5%

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

   if 2.6999999999999999e23 < u

   1. Initial program 84.5%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around 0 89.6%

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

      1. associate-*r/ 89.6%

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

      2. mul-1-neg 89.6%

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

   6. Simplified 89.6%

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

      1. *-commutative 89.6%

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

      2. clear-num 89.6%

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

      3. frac-2neg 89.6%

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

      4. frac-times 91.3%

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

      5. *-un-lft-identity 91.3%

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

      6. remove-double-neg 91.3%

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

   8. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{\frac{t1}{\frac{t1 + u}{v} \cdot \left(-u\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1600000000000:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq -9 \cdot 10^{-106}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 8 \cdot 10^{-130}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;u \leq 1.02 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{v}{\frac{t1 - u}{t1}}}{t1 + u}\\ \mathbf{elif}\;u \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{\left(-t1\right) - u}{v}}\\ \end{array} \]

Alternative 3: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{if}\;u \leq -300000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v (+ t1 u)) (/ (- t1) u))))
   (if (<= u -300000000000.0)
     t_1
     (if (<= u -1e-105)
       (/ v (- (* u -2.0) t1))
       (if (<= u -2.7e-151)
         (/ v (* (/ u t1) (- t1 u)))
         (if (<= u 1.15e+20) (- (/ v t1)) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / (t1 + u)) * (-t1 / u);
	double tmp;
	if (u <= -300000000000.0) {
		tmp = t_1;
	} else if (u <= -1e-105) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -2.7e-151) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 1.15e+20) {
		tmp = -(v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (v / (t1 + u)) * (-t1 / u)
    if (u <= (-300000000000.0d0)) then
        tmp = t_1
    else if (u <= (-1d-105)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if (u <= (-2.7d-151)) then
        tmp = v / ((u / t1) * (t1 - u))
    else if (u <= 1.15d+20) then
        tmp = -(v / t1)
    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)) * (-t1 / u);
	double tmp;
	if (u <= -300000000000.0) {
		tmp = t_1;
	} else if (u <= -1e-105) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -2.7e-151) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 1.15e+20) {
		tmp = -(v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / (t1 + u)) * (-t1 / u)
	tmp = 0
	if u <= -300000000000.0:
		tmp = t_1
	elif u <= -1e-105:
		tmp = v / ((u * -2.0) - t1)
	elif u <= -2.7e-151:
		tmp = v / ((u / t1) * (t1 - u))
	elif u <= 1.15e+20:
		tmp = -(v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u))
	tmp = 0.0
	if (u <= -300000000000.0)
		tmp = t_1;
	elseif (u <= -1e-105)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif (u <= -2.7e-151)
		tmp = Float64(v / Float64(Float64(u / t1) * Float64(t1 - u)));
	elseif (u <= 1.15e+20)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / (t1 + u)) * (-t1 / u);
	tmp = 0.0;
	if (u <= -300000000000.0)
		tmp = t_1;
	elseif (u <= -1e-105)
		tmp = v / ((u * -2.0) - t1);
	elseif (u <= -2.7e-151)
		tmp = v / ((u / t1) * (t1 - u));
	elseif (u <= 1.15e+20)
		tmp = -(v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -300000000000.0], t$95$1, If[LessEqual[u, -1e-105], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -2.7e-151], N[(v / N[(N[(u / t1), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.15e+20], (-N[(v / t1), $MachinePrecision]), t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -3e11 or 1.15e20 < u

   1. Initial program 80.3%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around 0 91.1%

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

      1. associate-*r/ 91.1%

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

      2. mul-1-neg 91.1%

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

   6. Simplified 91.1%

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

   if -3e11 < u < -9.99999999999999965e-106

   1. Initial program 82.0%

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

      1. associate-/r* 85.8%

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

      2. *-commutative 85.8%

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

      3. associate-/l* 99.7%

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

      4. associate-/l/ 99.7%

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

      5. +-commutative 99.7%

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

      6. remove-double-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. div-sub 99.7%

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

      9. sub-neg 99.7%

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

      10. *-inverses 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t1 around inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. unsub-neg 74.6%

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

      3. *-commutative 74.6%

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

   6. Simplified 74.6%

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

   if -9.99999999999999965e-106 < u < -2.70000000000000007e-151

   1. Initial program 61.5%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in t1 around 0 61.2%

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

      1. associate-*r/ 61.2%

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

      2. mul-1-neg 61.2%

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

   6. Simplified 61.2%

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

      1. associate-*r/ 61.3%

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

      2. frac-2neg 61.3%

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

      3. add-sqr-sqrt 10.5%

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

      4. sqrt-unprod 13.3%

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

      5. sqr-neg 13.3%

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

      6. sqrt-unprod 12.0%

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

      7. add-sqr-sqrt 12.7%

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

      8. distribute-lft-neg-out 12.7%

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

      9. distribute-frac-neg 12.7%

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

      10. clear-num 12.7%

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

      11. associate-*l/ 12.7%

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

      12. *-un-lft-identity 12.7%

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

      13. add-sqr-sqrt 0.7%

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

      14. sqrt-unprod 22.6%

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

      15. sqr-neg 22.6%

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

      16. sqrt-unprod 50.6%

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

      17. add-sqr-sqrt 61.0%

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

      18. distribute-neg-in 61.0%

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

   8. Applied egg-rr 61.3%

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

      1. associate-/l/ 80.2%

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

      2. *-commutative 80.2%

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

   10. Simplified 80.2%

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

   if -2.70000000000000007e-151 < u < 1.15e20

   1. Initial program 65.4%

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

      1. times-frac 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t1 around inf 79.7%

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

      1. associate-*r/ 79.7%

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

      2. neg-mul-1 79.7%

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

   6. Simplified 79.7%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -300000000000:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 4: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -265000000000:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq -9 \cdot 10^{-106}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -1.9 \cdot 10^{-151}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{+21}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{\left(-t1\right) - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -265000000000.0)
   (* (/ v (+ t1 u)) (/ (- t1) u))
   (if (<= u -9e-106)
     (/ v (- (* u -2.0) t1))
     (if (<= u -1.9e-151)
       (/ v (* (/ u t1) (- t1 u)))
       (if (<= u 4.5e+21) (- (/ v t1)) (/ t1 (* u (/ (- (- t1) u) v))))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -265000000000.0) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else if (u <= -9e-106) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -1.9e-151) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 4.5e+21) {
		tmp = -(v / t1);
	} else {
		tmp = 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 <= (-265000000000.0d0)) then
        tmp = (v / (t1 + u)) * (-t1 / u)
    else if (u <= (-9d-106)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if (u <= (-1.9d-151)) then
        tmp = v / ((u / t1) * (t1 - u))
    else if (u <= 4.5d+21) then
        tmp = -(v / t1)
    else
        tmp = 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 <= -265000000000.0) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else if (u <= -9e-106) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -1.9e-151) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 4.5e+21) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * ((-t1 - u) / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -265000000000.0:
		tmp = (v / (t1 + u)) * (-t1 / u)
	elif u <= -9e-106:
		tmp = v / ((u * -2.0) - t1)
	elif u <= -1.9e-151:
		tmp = v / ((u / t1) * (t1 - u))
	elif u <= 4.5e+21:
		tmp = -(v / t1)
	else:
		tmp = t1 / (u * ((-t1 - u) / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -265000000000.0)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	elseif (u <= -9e-106)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif (u <= -1.9e-151)
		tmp = Float64(v / Float64(Float64(u / t1) * Float64(t1 - u)));
	elseif (u <= 4.5e+21)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(Float64(-t1) - u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -265000000000.0)
		tmp = (v / (t1 + u)) * (-t1 / u);
	elseif (u <= -9e-106)
		tmp = v / ((u * -2.0) - t1);
	elseif (u <= -1.9e-151)
		tmp = v / ((u / t1) * (t1 - u));
	elseif (u <= 4.5e+21)
		tmp = -(v / t1);
	else
		tmp = t1 / (u * ((-t1 - u) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -265000000000.0], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -9e-106], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1.9e-151], N[(v / N[(N[(u / t1), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.5e+21], (-N[(v / t1), $MachinePrecision]), N[(t1 / N[(u * N[(N[((-t1) - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -2.65e11

   1. Initial program 76.3%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around 0 92.5%

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

      1. associate-*r/ 92.5%

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

      2. mul-1-neg 92.5%

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

   6. Simplified 92.5%

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

   if -2.65e11 < u < -8.99999999999999911e-106

   1. Initial program 82.0%

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

      1. associate-/r* 85.8%

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

      2. *-commutative 85.8%

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

      3. associate-/l* 99.7%

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

      4. associate-/l/ 99.7%

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

      5. +-commutative 99.7%

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

      6. remove-double-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. div-sub 99.7%

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

      9. sub-neg 99.7%

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

      10. *-inverses 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t1 around inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. unsub-neg 74.6%

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

      3. *-commutative 74.6%

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

   6. Simplified 74.6%

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

   if -8.99999999999999911e-106 < u < -1.89999999999999985e-151

   1. Initial program 61.5%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in t1 around 0 61.2%

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

      1. associate-*r/ 61.2%

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

      2. mul-1-neg 61.2%

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

   6. Simplified 61.2%

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

      1. associate-*r/ 61.3%

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

      2. frac-2neg 61.3%

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

      3. add-sqr-sqrt 10.5%

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

      4. sqrt-unprod 13.3%

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

      5. sqr-neg 13.3%

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

      6. sqrt-unprod 12.0%

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

      7. add-sqr-sqrt 12.7%

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

      8. distribute-lft-neg-out 12.7%

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

      9. distribute-frac-neg 12.7%

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

      10. clear-num 12.7%

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

      11. associate-*l/ 12.7%

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

      12. *-un-lft-identity 12.7%

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

      13. add-sqr-sqrt 0.7%

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

      14. sqrt-unprod 22.6%

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

      15. sqr-neg 22.6%

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

      16. sqrt-unprod 50.6%

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

      17. add-sqr-sqrt 61.0%

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

      18. distribute-neg-in 61.0%

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

   8. Applied egg-rr 61.3%

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

      1. associate-/l/ 80.2%

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

      2. *-commutative 80.2%

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

   10. Simplified 80.2%

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

   if -1.89999999999999985e-151 < u < 4.5e21

   1. Initial program 65.4%

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

      1. times-frac 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t1 around inf 79.7%

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

      1. associate-*r/ 79.7%

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

      2. neg-mul-1 79.7%

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

   6. Simplified 79.7%

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

   if 4.5e21 < u

   1. Initial program 84.5%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around 0 89.6%

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

      1. associate-*r/ 89.6%

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

      2. mul-1-neg 89.6%

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

   6. Simplified 89.6%

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

      1. *-commutative 89.6%

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

      2. clear-num 89.6%

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

      3. frac-2neg 89.6%

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

      4. frac-times 91.3%

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

      5. *-un-lft-identity 91.3%

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

      6. remove-double-neg 91.3%

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

   8. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{\frac{t1}{\frac{t1 + u}{v} \cdot \left(-u\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -265000000000:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq -9 \cdot 10^{-106}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -1.9 \cdot 10^{-151}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{+21}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{\left(-t1\right) - u}{v}}\\ \end{array} \]

Alternative 5: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -5.3 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -1.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -6 \cdot 10^{-123} \lor \neg \left(u \leq 2 \cdot 10^{+21}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- t1) u) (/ v u))))
   (if (<= u -5.3e+43)
     t_1
     (if (<= u -1.3e-105)
       (/ v (- (* u -2.0) t1))
       (if (or (<= u -6e-123) (not (<= u 2e+21))) t_1 (- (/ v t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -5.3e+43) {
		tmp = t_1;
	} else if (u <= -1.3e-105) {
		tmp = v / ((u * -2.0) - t1);
	} else if ((u <= -6e-123) || !(u <= 2e+21)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (-t1 / u) * (v / u)
    if (u <= (-5.3d+43)) then
        tmp = t_1
    else if (u <= (-1.3d-105)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if ((u <= (-6d-123)) .or. (.not. (u <= 2d+21))) then
        tmp = t_1
    else
        tmp = -(v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -5.3e+43) {
		tmp = t_1;
	} else if (u <= -1.3e-105) {
		tmp = v / ((u * -2.0) - t1);
	} else if ((u <= -6e-123) || !(u <= 2e+21)) {
		tmp = t_1;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 / u) * (v / u)
	tmp = 0
	if u <= -5.3e+43:
		tmp = t_1
	elif u <= -1.3e-105:
		tmp = v / ((u * -2.0) - t1)
	elif (u <= -6e-123) or not (u <= 2e+21):
		tmp = t_1
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) / u) * Float64(v / u))
	tmp = 0.0
	if (u <= -5.3e+43)
		tmp = t_1;
	elseif (u <= -1.3e-105)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif ((u <= -6e-123) || !(u <= 2e+21))
		tmp = t_1;
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 / u) * (v / u);
	tmp = 0.0;
	if (u <= -5.3e+43)
		tmp = t_1;
	elseif (u <= -1.3e-105)
		tmp = v / ((u * -2.0) - t1);
	elseif ((u <= -6e-123) || ~((u <= 2e+21)))
		tmp = t_1;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -5.3e+43], t$95$1, If[LessEqual[u, -1.3e-105], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -6e-123], N[Not[LessEqual[u, 2e+21]], $MachinePrecision]], t$95$1, (-N[(v / t1), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.2999999999999999e43 or -1.2999999999999999e-105 < u < -5.99999999999999968e-123 or 2e21 < u

   1. Initial program 79.4%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around 0 91.9%

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

      1. associate-*r/ 91.9%

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

      2. mul-1-neg 91.9%

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

   6. Simplified 91.9%

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

   7. Taylor expanded in t1 around 0 86.7%

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

   if -5.2999999999999999e43 < u < -1.2999999999999999e-105

   1. Initial program 82.3%

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

      1. associate-/r* 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-/l* 99.6%

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

      4. associate-/l/ 99.7%

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

      5. +-commutative 99.7%

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

      6. remove-double-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. div-sub 99.7%

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

      9. sub-neg 99.7%

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

      10. *-inverses 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t1 around inf 70.7%

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

      1. mul-1-neg 70.7%

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

      2. unsub-neg 70.7%

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

      3. *-commutative 70.7%

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

   6. Simplified 70.7%

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

   if -5.99999999999999968e-123 < u < 2e21

   1. Initial program 65.2%

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

      1. times-frac 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in t1 around inf 78.0%

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

      1. associate-*r/ 78.0%

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

      2. neg-mul-1 78.0%

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

   6. Simplified 78.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.3 \cdot 10^{+43}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq -1.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -6 \cdot 10^{-123} \lor \neg \left(u \leq 2 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 6: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -4.3 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -9 \cdot 10^{-106}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -6 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 12000000:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- t1) u) (/ v u))))
   (if (<= u -4.3e+35)
     t_1
     (if (<= u -9e-106)
       (/ v (- (* u -2.0) t1))
       (if (<= u -6e-123)
         t_1
         (if (<= u 12000000.0) (- (/ v t1)) (/ t1 (* u (/ (- u) v)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -4.3e+35) {
		tmp = t_1;
	} else if (u <= -9e-106) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -6e-123) {
		tmp = t_1;
	} else if (u <= 12000000.0) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * (-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 = (-t1 / u) * (v / u)
    if (u <= (-4.3d+35)) then
        tmp = t_1
    else if (u <= (-9d-106)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if (u <= (-6d-123)) then
        tmp = t_1
    else if (u <= 12000000.0d0) then
        tmp = -(v / t1)
    else
        tmp = t1 / (u * (-u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -4.3e+35) {
		tmp = t_1;
	} else if (u <= -9e-106) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -6e-123) {
		tmp = t_1;
	} else if (u <= 12000000.0) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * (-u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 / u) * (v / u)
	tmp = 0
	if u <= -4.3e+35:
		tmp = t_1
	elif u <= -9e-106:
		tmp = v / ((u * -2.0) - t1)
	elif u <= -6e-123:
		tmp = t_1
	elif u <= 12000000.0:
		tmp = -(v / t1)
	else:
		tmp = t1 / (u * (-u / v))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) / u) * Float64(v / u))
	tmp = 0.0
	if (u <= -4.3e+35)
		tmp = t_1;
	elseif (u <= -9e-106)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif (u <= -6e-123)
		tmp = t_1;
	elseif (u <= 12000000.0)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(-u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 / u) * (v / u);
	tmp = 0.0;
	if (u <= -4.3e+35)
		tmp = t_1;
	elseif (u <= -9e-106)
		tmp = v / ((u * -2.0) - t1);
	elseif (u <= -6e-123)
		tmp = t_1;
	elseif (u <= 12000000.0)
		tmp = -(v / t1);
	else
		tmp = t1 / (u * (-u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4.3e+35], t$95$1, If[LessEqual[u, -9e-106], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -6e-123], t$95$1, If[LessEqual[u, 12000000.0], (-N[(v / t1), $MachinePrecision]), N[(t1 / N[(u * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -4.2999999999999998e35 or -8.99999999999999911e-106 < u < -5.99999999999999968e-123

   1. Initial program 74.4%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around 0 94.0%

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

      1. associate-*r/ 94.0%

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

      2. mul-1-neg 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in t1 around 0 90.6%

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

   if -4.2999999999999998e35 < u < -8.99999999999999911e-106

   1. Initial program 82.3%

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

      1. associate-/r* 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-/l* 99.6%

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

      4. associate-/l/ 99.7%

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

      5. +-commutative 99.7%

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

      6. remove-double-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. div-sub 99.7%

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

      9. sub-neg 99.7%

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

      10. *-inverses 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t1 around inf 70.7%

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

      1. mul-1-neg 70.7%

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

      2. unsub-neg 70.7%

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

      3. *-commutative 70.7%

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

   6. Simplified 70.7%

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

   if -5.99999999999999968e-123 < u < 1.2e7

   1. Initial program 63.9%

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

      1. times-frac 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in t1 around inf 79.0%

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

      1. associate-*r/ 79.0%

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

      2. neg-mul-1 79.0%

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

   6. Simplified 79.0%

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

   if 1.2e7 < u

   1. Initial program 85.5%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around 0 87.1%

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

      1. associate-*r/ 87.1%

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

      2. mul-1-neg 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in t1 around 0 80.6%

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

      1. *-commutative 80.6%

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

      2. clear-num 80.6%

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

      3. frac-2neg 80.6%

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

      4. frac-times 85.5%

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

      5. *-un-lft-identity 85.5%

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

      6. remove-double-neg 85.5%

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

   9. Applied egg-rr 85.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{+35}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq -9 \cdot 10^{-106}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -6 \cdot 10^{-123}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 12000000:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \end{array} \]

Alternative 7: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.55 \cdot 10^{+43}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq -9 \cdot 10^{-106}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 16500000:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.55e+43)
   (* (/ (- t1) u) (/ v u))
   (if (<= u -9e-106)
     (/ v (- (* u -2.0) t1))
     (if (<= u -2.7e-151)
       (/ v (* (/ u t1) (- t1 u)))
       (if (<= u 16500000.0) (- (/ v t1)) (/ t1 (* u (/ (- u) v))))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.55e+43) {
		tmp = (-t1 / u) * (v / u);
	} else if (u <= -9e-106) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -2.7e-151) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 16500000.0) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * (-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 <= (-2.55d+43)) then
        tmp = (-t1 / u) * (v / u)
    else if (u <= (-9d-106)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if (u <= (-2.7d-151)) then
        tmp = v / ((u / t1) * (t1 - u))
    else if (u <= 16500000.0d0) then
        tmp = -(v / t1)
    else
        tmp = t1 / (u * (-u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.55e+43) {
		tmp = (-t1 / u) * (v / u);
	} else if (u <= -9e-106) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -2.7e-151) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 16500000.0) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * (-u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.55e+43:
		tmp = (-t1 / u) * (v / u)
	elif u <= -9e-106:
		tmp = v / ((u * -2.0) - t1)
	elif u <= -2.7e-151:
		tmp = v / ((u / t1) * (t1 - u))
	elif u <= 16500000.0:
		tmp = -(v / t1)
	else:
		tmp = t1 / (u * (-u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.55e+43)
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	elseif (u <= -9e-106)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif (u <= -2.7e-151)
		tmp = Float64(v / Float64(Float64(u / t1) * Float64(t1 - u)));
	elseif (u <= 16500000.0)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(-u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.55e+43)
		tmp = (-t1 / u) * (v / u);
	elseif (u <= -9e-106)
		tmp = v / ((u * -2.0) - t1);
	elseif (u <= -2.7e-151)
		tmp = v / ((u / t1) * (t1 - u));
	elseif (u <= 16500000.0)
		tmp = -(v / t1);
	else
		tmp = t1 / (u * (-u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.55e+43], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -9e-106], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -2.7e-151], N[(v / N[(N[(u / t1), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 16500000.0], (-N[(v / t1), $MachinePrecision]), N[(t1 / N[(u * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -2.54999999999999997e43

   1. Initial program 75.5%

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

      1. times-frac 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in t1 around 0 93.5%

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

      1. associate-*r/ 93.5%

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

      2. mul-1-neg 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in t1 around 0 89.8%

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

   if -2.54999999999999997e43 < u < -8.99999999999999911e-106

   1. Initial program 82.3%

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

      1. associate-/r* 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-/l* 99.6%

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

      4. associate-/l/ 99.7%

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

      5. +-commutative 99.7%

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

      6. remove-double-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. div-sub 99.7%

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

      9. sub-neg 99.7%

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

      10. *-inverses 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t1 around inf 70.7%

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

      1. mul-1-neg 70.7%

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

      2. unsub-neg 70.7%

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

      3. *-commutative 70.7%

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

   6. Simplified 70.7%

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

   if -8.99999999999999911e-106 < u < -2.70000000000000007e-151

   1. Initial program 61.5%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in t1 around 0 61.2%

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

      1. associate-*r/ 61.2%

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

      2. mul-1-neg 61.2%

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

   6. Simplified 61.2%

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

      1. associate-*r/ 61.3%

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

      2. frac-2neg 61.3%

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

      3. add-sqr-sqrt 10.5%

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

      4. sqrt-unprod 13.3%

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

      5. sqr-neg 13.3%

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

      6. sqrt-unprod 12.0%

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

      7. add-sqr-sqrt 12.7%

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

      8. distribute-lft-neg-out 12.7%

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

      9. distribute-frac-neg 12.7%

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

      10. clear-num 12.7%

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

      11. associate-*l/ 12.7%

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

      12. *-un-lft-identity 12.7%

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

      13. add-sqr-sqrt 0.7%

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

      14. sqrt-unprod 22.6%

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

      15. sqr-neg 22.6%

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

      16. sqrt-unprod 50.6%

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

      17. add-sqr-sqrt 61.0%

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

      18. distribute-neg-in 61.0%

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

   8. Applied egg-rr 61.3%

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

      1. associate-/l/ 80.2%

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

      2. *-commutative 80.2%

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

   10. Simplified 80.2%

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

   if -2.70000000000000007e-151 < u < 1.65e7

   1. Initial program 64.1%

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

      1. times-frac 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in t1 around inf 80.8%

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

      1. associate-*r/ 80.8%

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

      2. neg-mul-1 80.8%

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

   6. Simplified 80.8%

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

   if 1.65e7 < u

   1. Initial program 85.5%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around 0 87.1%

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

      1. associate-*r/ 87.1%

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

      2. mul-1-neg 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in t1 around 0 80.6%

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

      1. *-commutative 80.6%

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

      2. clear-num 80.6%

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

      3. frac-2neg 80.6%

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

      4. frac-times 85.5%

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

      5. *-un-lft-identity 85.5%

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

      6. remove-double-neg 85.5%

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

   9. Applied egg-rr 85.5%

      \[\leadsto \color{blue}{\frac{t1}{\frac{u}{v} \cdot \left(-u\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.55 \cdot 10^{+43}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq -9 \cdot 10^{-106}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 16500000:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \end{array} \]

Alternative 8: 67.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+68} \lor \neg \left(u \leq 4.7 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5e+68) (not (<= u 4.7e+160)))
   (/ t1 (* u (/ u v)))
   (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5e+68) || !(u <= 4.7e+160)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5d+68)) .or. (.not. (u <= 4.7d+160))) then
        tmp = t1 / (u * (u / v))
    else
        tmp = -(v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5e+68) || !(u <= 4.7e+160)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5e+68) or not (u <= 4.7e+160):
		tmp = t1 / (u * (u / v))
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5e+68) || !(u <= 4.7e+160))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5e+68) || ~((u <= 4.7e+160)))
		tmp = t1 / (u * (u / v));
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5e+68], N[Not[LessEqual[u, 4.7e+160]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -5.0000000000000004e68 or 4.6999999999999997e160 < u

   1. Initial program 77.3%

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

      1. times-frac 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in t1 around 0 96.2%

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

      1. associate-*r/ 96.2%

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

      2. mul-1-neg 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in t1 around 0 93.7%

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

      1. *-commutative 93.7%

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

      2. clear-num 93.7%

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

      3. frac-times 92.8%

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

      4. *-un-lft-identity 92.8%

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

      5. add-sqr-sqrt 46.6%

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

      6. sqrt-unprod 64.5%

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

      7. sqr-neg 64.5%

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

      8. sqrt-unprod 34.6%

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

      9. add-sqr-sqrt 74.9%

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

   9. Applied egg-rr 74.9%

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

   if -5.0000000000000004e68 < u < 4.6999999999999997e160

   1. Initial program 72.2%

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

      1. times-frac 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in t1 around inf 66.4%

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

      1. associate-*r/ 66.4%

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

      2. neg-mul-1 66.4%

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

   6. Simplified 66.4%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+68} \lor \neg \left(u \leq 4.7 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 9: 68.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.4 \cdot 10^{+73} \lor \neg \left(u \leq 5 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -9.4e+73) (not (<= u 5e+160)))
   (/ t1 (* u (/ u v)))
   (/ v (- (* u -2.0) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.4e+73) || !(u <= 5e+160)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / ((u * -2.0) - 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 <= (-9.4d+73)) .or. (.not. (u <= 5d+160))) then
        tmp = t1 / (u * (u / v))
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.4e+73) || !(u <= 5e+160)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -9.4e+73) or not (u <= 5e+160):
		tmp = t1 / (u * (u / v))
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -9.4e+73) || !(u <= 5e+160))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -9.4e+73) || ~((u <= 5e+160)))
		tmp = t1 / (u * (u / v));
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -9.4e+73], N[Not[LessEqual[u, 5e+160]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -9.4000000000000004e73 or 5.0000000000000002e160 < u

   1. Initial program 77.0%

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

      1. times-frac 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in t1 around 0 96.1%

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

      1. associate-*r/ 96.1%

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

      2. mul-1-neg 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in t1 around 0 93.6%

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

      1. *-commutative 93.6%

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

      2. clear-num 93.6%

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

      3. frac-times 92.7%

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

      4. *-un-lft-identity 92.7%

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

      5. add-sqr-sqrt 45.9%

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

      6. sqrt-unprod 64.0%

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

      7. sqr-neg 64.0%

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

      8. sqrt-unprod 35.1%

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

      9. add-sqr-sqrt 74.5%

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

   9. Applied egg-rr 74.5%

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

   if -9.4000000000000004e73 < u < 5.0000000000000002e160

   1. Initial program 72.4%

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

      1. associate-/r* 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/l* 98.0%

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

      4. associate-/l/ 98.2%

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

      5. +-commutative 98.2%

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

      6. remove-double-neg 98.2%

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

      7. unsub-neg 98.2%

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

      8. div-sub 98.2%

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

      9. sub-neg 98.2%

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

      10. *-inverses 98.2%

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

      11. metadata-eval 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around inf 66.6%

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

      1. mul-1-neg 66.6%

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

      2. unsub-neg 66.6%

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

      3. *-commutative 66.6%

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

   6. Simplified 66.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.4 \cdot 10^{+73} \lor \neg \left(u \leq 5 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 10: 55.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -9.2e+73) (* (/ v u) -0.5) (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9.2e+73) {
		tmp = (v / u) * -0.5;
	} 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 <= (-9.2d+73)) then
        tmp = (v / u) * (-0.5d0)
    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 <= -9.2e+73) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -9.2e+73:
		tmp = (v / u) * -0.5
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -9.2e+73)
		tmp = Float64(Float64(v / u) * -0.5);
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -9.2e+73)
		tmp = (v / u) * -0.5;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -9.2e+73], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -9.199999999999999e73

   1. Initial program 74.1%

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

      1. associate-/r* 86.4%

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

      2. *-commutative 86.4%

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

      3. associate-/l* 97.1%

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

      4. associate-/l/ 82.2%

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

      5. +-commutative 82.2%

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

      6. remove-double-neg 82.2%

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

      7. unsub-neg 82.2%

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

      8. div-sub 82.3%

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

      9. sub-neg 82.3%

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

      10. *-inverses 82.3%

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

      11. metadata-eval 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in t1 around inf 38.8%

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

      1. mul-1-neg 38.8%

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

      2. unsub-neg 38.8%

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

      3. *-commutative 38.8%

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

   6. Simplified 38.8%

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

   7. Taylor expanded in u around inf 35.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   8. Step-by-step derivation

      1. *-commutative 35.1%

         \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]

   9. Simplified 35.1%

      \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]

   if -9.199999999999999e73 < u

   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. times-frac 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t1 around inf 60.1%

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

      1. associate-*r/ 60.1%

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

      2. neg-mul-1 60.1%

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

   6. Simplified 60.1%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 11: 55.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.2 \cdot 10^{+73}:\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -9.2e+73) (- (/ v u)) (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9.2e+73) {
		tmp = -(v / u);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-9.2d+73)) then
        tmp = -(v / u)
    else
        tmp = -(v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9.2e+73) {
		tmp = -(v / u);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -9.2e+73:
		tmp = -(v / u)
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -9.2e+73)
		tmp = Float64(-Float64(v / u));
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -9.2e+73)
		tmp = -(v / u);
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -9.2e+73], (-N[(v / u), $MachinePrecision]), (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -9.199999999999999e73

   1. Initial program 74.1%

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

      1. times-frac 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in t1 around 0 94.1%

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

      1. associate-*r/ 94.1%

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

      2. mul-1-neg 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in t1 around inf 35.0%

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

      1. associate-*r/ 35.0%

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

      2. neg-mul-1 35.0%

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

   9. Simplified 35.0%

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

   if -9.199999999999999e73 < u

   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. times-frac 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t1 around inf 60.1%

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

      1. associate-*r/ 60.1%

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

      2. neg-mul-1 60.1%

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

   6. Simplified 60.1%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.2 \cdot 10^{+73}:\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 12: 53.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. times-frac 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in t1 around inf 52.1%

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

   1. associate-*r/ 52.1%

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

   2. neg-mul-1 52.1%

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

6. Simplified 52.1%

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

7. Final simplification 52.1%

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

Alternative 13: 13.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. times-frac 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in t1 around inf 52.3%

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

5. Taylor expanded in u around inf 13.2%

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

6. Final simplification 13.2%

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

Reproduce

herbie shell --seed 2023332 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))