Rosa's DopplerBench

Percentage Accurate: 73.2% → 98.2%

Time: 10.7s

Alternatives: 16

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{v}{-1 - \frac{u}{t1}}}{u + t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (/ v (- -1.0 (/ u t1))) (+ u t1)))

C

double code(double u, double v, double t1) {
	return (v / (-1.0 - (u / t1))) / (u + t1);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v / ((-1.0d0) - (u / t1))) / (u + t1)
end function

Java

public static double code(double u, double v, double t1) {
	return (v / (-1.0 - (u / t1))) / (u + t1);
}

Python

def code(u, v, t1):
	return (v / (-1.0 - (u / t1))) / (u + t1)

Julia

function code(u, v, t1)
	return Float64(Float64(v / Float64(-1.0 - Float64(u / t1))) / Float64(u + t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v / (-1.0 - (u / t1))) / (u + t1);
end

Wolfram

code[u_, v_, t1_] := N[(N[(v / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. 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. Step-by-step derivation

   1. associate-*r/ 97.7%

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

   2. clear-num 97.5%

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

   3. associate-*l/ 97.5%

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

   4. *-un-lft-identity 97.5%

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

   5. frac-2neg 97.5%

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

   6. distribute-neg-in 97.5%

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

   7. add-sqr-sqrt 48.9%

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

   8. sqrt-unprod 75.4%

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

   9. sqr-neg 75.4%

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

   10. sqrt-unprod 34.5%

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

   11. add-sqr-sqrt 65.8%

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

   12. sub-neg 65.8%

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

   13. remove-double-neg 65.8%

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

5. Applied egg-rr 65.8%

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

   1. frac-2neg 65.8%

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

   2. distribute-frac-neg 65.8%

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

   3. add-sqr-sqrt 31.3%

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

   4. sqrt-unprod 45.1%

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

   5. sqr-neg 45.1%

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

   6. sqrt-unprod 35.6%

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

   7. add-sqr-sqrt 74.3%

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

   8. neg-sub0 74.3%

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

   9. div-sub 74.3%

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

   10. *-inverses 74.3%

       \[\leadsto \frac{\frac{v}{0 - \left(\color{blue}{1} - \frac{u}{t1}\right)}}{t1 + u} \]

   11. sub-neg 74.3%

       \[\leadsto \frac{\frac{v}{0 - \color{blue}{\left(1 + \left(-\frac{u}{t1}\right)\right)}}}{t1 + u} \]

   12. add-sqr-sqrt 35.7%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \left(-\frac{u}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}\right)\right)}}{t1 + u} \]

   13. sqrt-unprod 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \left(-\frac{u}{\color{blue}{\sqrt{t1 \cdot t1}}}\right)\right)}}{t1 + u} \]

   14. sqr-neg 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \left(-\frac{u}{\sqrt{\color{blue}{\left(-t1\right) \cdot \left(-t1\right)}}}\right)\right)}}{t1 + u} \]

   15. sqrt-unprod 49.1%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \left(-\frac{u}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}\right)\right)}}{t1 + u} \]

   16. add-sqr-sqrt 97.5%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \left(-\frac{u}{\color{blue}{-t1}}\right)\right)}}{t1 + u} \]

   17. distribute-frac-neg 97.5%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \color{blue}{\frac{-u}{-t1}}\right)}}{t1 + u} \]

   18. frac-2neg 97.5%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \color{blue}{\frac{u}{t1}}\right)}}{t1 + u} \]

7. Applied egg-rr 97.5%

   \[\leadsto \frac{\frac{v}{\color{blue}{0 - \left(1 + \frac{u}{t1}\right)}}}{t1 + u} \]
8. Step-by-step derivation

   1. associate--r+ 97.5%

      \[\leadsto \frac{\frac{v}{\color{blue}{\left(0 - 1\right) - \frac{u}{t1}}}}{t1 + u} \]

   2. metadata-eval 97.5%

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

9. Simplified 97.5%

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

10. Final simplification 97.5%

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

Alternative 2: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.8 \cdot 10^{-88} \lor \neg \left(t1 \leq 3 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{v \cdot t1}{u}}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.8e-88) (not (<= t1 3e-8)))
   (/ v (- (* u -2.0) t1))
   (- (/ (/ (* v t1) u) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.8e-88) || !(t1 <= 3e-8)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = -(((v * t1) / u) / (u + t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4.8d-88)) .or. (.not. (t1 <= 3d-8))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = -(((v * t1) / u) / (u + t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.8e-88) || !(t1 <= 3e-8)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = -(((v * t1) / u) / (u + t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.8e-88) or not (t1 <= 3e-8):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = -(((v * t1) / u) / (u + t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.8e-88) || !(t1 <= 3e-8))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(-Float64(Float64(Float64(v * t1) / u) / Float64(u + t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.8e-88) || ~((t1 <= 3e-8)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = -(((v * t1) / u) / (u + t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.8e-88], N[Not[LessEqual[t1, 3e-8]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[(v * t1), $MachinePrecision] / u), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.7999999999999999e-88 or 2.99999999999999973e-8 < t1

   1. Initial program 73.7%

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

      1. associate-/r* 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 94.7%

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

      5. +-commutative 94.7%

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

      6. remove-double-neg 94.7%

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

      7. unsub-neg 94.7%

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

      8. div-sub 94.7%

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

      9. sub-neg 94.7%

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

      10. *-inverses 94.7%

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

      11. metadata-eval 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in t1 around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. unsub-neg 85.8%

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

      3. *-commutative 85.8%

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

   6. Simplified 85.8%

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

   if -4.7999999999999999e-88 < t1 < 2.99999999999999973e-8

   1. Initial program 84.6%

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

      1. times-frac 93.8%

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

   3. Simplified 93.8%

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

      1. associate-*r/ 94.9%

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

      2. clear-num 94.5%

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

      3. associate-*l/ 94.5%

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

      4. *-un-lft-identity 94.5%

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

      5. frac-2neg 94.5%

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

      6. distribute-neg-in 94.5%

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

      7. add-sqr-sqrt 39.5%

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

      8. sqrt-unprod 82.9%

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

      9. sqr-neg 82.9%

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

      10. sqrt-unprod 47.6%

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

      11. add-sqr-sqrt 81.0%

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

      12. sub-neg 81.0%

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

      13. remove-double-neg 81.0%

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

   5. Applied egg-rr 81.0%

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

   6. Taylor expanded in t1 around 0 82.4%

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

      1. mul-1-neg 82.4%

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

      2. *-commutative 82.4%

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

   8. Simplified 82.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.8 \cdot 10^{-88} \lor \neg \left(t1 \leq 3 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{v \cdot t1}{u}}{u + t1}\\ \end{array} \]

Alternative 3: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.95 \cdot 10^{-88} \lor \neg \left(t1 \leq 255000000\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \left(-\frac{v}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.95e-88) (not (<= t1 255000000.0)))
   (/ v (- (* u -2.0) t1))
   (* (/ t1 u) (- (/ v u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.95e-88) || !(t1 <= 255000000.0)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 / u) * -(v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3.95d-88)) .or. (.not. (t1 <= 255000000.0d0))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (t1 / u) * -(v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.95e-88) || !(t1 <= 255000000.0)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 / u) * -(v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.95e-88) or not (t1 <= 255000000.0):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (t1 / u) * -(v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.95e-88) || !(t1 <= 255000000.0))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(t1 / u) * Float64(-Float64(v / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.95e-88) || ~((t1 <= 255000000.0)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (t1 / u) * -(v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.95e-88], N[Not[LessEqual[t1, 255000000.0]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * (-N[(v / u), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.94999999999999983e-88 or 2.55e8 < t1

   1. Initial program 73.8%

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

      1. associate-/r* 79.9%

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

      2. *-commutative 79.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 95.3%

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

      5. +-commutative 95.3%

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

      6. remove-double-neg 95.3%

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

      7. unsub-neg 95.3%

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

      8. div-sub 95.3%

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

      9. sub-neg 95.3%

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

      10. *-inverses 95.3%

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

      11. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in t1 around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. unsub-neg 86.2%

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

      3. *-commutative 86.2%

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

   6. Simplified 86.2%

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

   if -3.94999999999999983e-88 < t1 < 2.55e8

   1. Initial program 84.1%

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

      1. times-frac 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in t1 around 0 80.2%

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

   5. Taylor expanded in t1 around 0 80.4%

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

      1. associate-*r/ 80.4%

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

      2. mul-1-neg 80.4%

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

   7. Simplified 80.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.95 \cdot 10^{-88} \lor \neg \left(t1 \leq 255000000\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \left(-\frac{v}{u}\right)\\ \end{array} \]

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.1 \cdot 10^{-88} \lor \neg \left(t1 \leq 6500000\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.1e-88) (not (<= t1 6500000.0)))
   (/ v (- (* u -2.0) t1))
   (/ (* v (/ t1 u)) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.1e-88) || !(t1 <= 6500000.0)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.1d-88)) .or. (.not. (t1 <= 6500000.0d0))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (v * (t1 / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.1e-88) || !(t1 <= 6500000.0)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.1e-88) or not (t1 <= 6500000.0):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (v * (t1 / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.1e-88) || !(t1 <= 6500000.0))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.1e-88) || ~((t1 <= 6500000.0)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (v * (t1 / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.1e-88], N[Not[LessEqual[t1, 6500000.0]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.10000000000000046e-88 or 6.5e6 < t1

   1. Initial program 73.8%

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

      1. associate-/r* 79.9%

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

      2. *-commutative 79.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 95.3%

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

      5. +-commutative 95.3%

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

      6. remove-double-neg 95.3%

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

      7. unsub-neg 95.3%

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

      8. div-sub 95.3%

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

      9. sub-neg 95.3%

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

      10. *-inverses 95.3%

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

      11. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in t1 around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. unsub-neg 86.2%

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

      3. *-commutative 86.2%

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

   6. Simplified 86.2%

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

   if -5.10000000000000046e-88 < t1 < 6.5e6

   1. Initial program 84.1%

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

      1. times-frac 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in t1 around 0 80.2%

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

   5. Taylor expanded in t1 around 0 80.4%

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

      1. associate-*r/ 80.4%

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

      2. mul-1-neg 80.4%

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

   7. Simplified 80.4%

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

      1. *-commutative 80.4%

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

      2. frac-2neg 80.4%

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

      3. remove-double-neg 80.4%

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

      4. associate-*r/ 80.9%

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

      5. add-sqr-sqrt 47.3%

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

      6. sqrt-unprod 57.3%

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

      7. sqr-neg 57.3%

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

      8. sqrt-unprod 18.8%

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

      9. add-sqr-sqrt 44.0%

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

      10. *-commutative 44.0%

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

      11. associate-*r/ 44.2%

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

      12. associate-*l/ 44.1%

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

      13. *-commutative 44.1%

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

      14. add-sqr-sqrt 18.8%

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

      15. sqrt-unprod 57.4%

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

      16. sqr-neg 57.4%

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

      17. sqrt-unprod 48.1%

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

      18. add-sqr-sqrt 81.5%

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

   9. Applied egg-rr 81.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.1 \cdot 10^{-88} \lor \neg \left(t1 \leq 6500000\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 5: 65.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{-176} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3e-176) (not (<= t1 3.1e-83)))
   (/ (- v) (- t1 u))
   (/ t1 (/ u (/ v u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e-176) || !(t1 <= 3.1e-83)) {
		tmp = -v / (t1 - u);
	} else {
		tmp = t1 / (u / (v / u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3d-176)) .or. (.not. (t1 <= 3.1d-83))) then
        tmp = -v / (t1 - u)
    else
        tmp = t1 / (u / (v / u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e-176) || !(t1 <= 3.1e-83)) {
		tmp = -v / (t1 - u);
	} else {
		tmp = t1 / (u / (v / u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3e-176) or not (t1 <= 3.1e-83):
		tmp = -v / (t1 - u)
	else:
		tmp = t1 / (u / (v / u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3e-176) || !(t1 <= 3.1e-83))
		tmp = Float64(Float64(-v) / Float64(t1 - u));
	else
		tmp = Float64(t1 / Float64(u / Float64(v / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3e-176) || ~((t1 <= 3.1e-83)))
		tmp = -v / (t1 - u);
	else
		tmp = t1 / (u / (v / u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3e-176], N[Not[LessEqual[t1, 3.1e-83]], $MachinePrecision]], N[((-v) / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3e-176 or 3.09999999999999992e-83 < t1

   1. Initial program 76.1%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. frac-2neg 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. add-sqr-sqrt 54.0%

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

      8. sqrt-unprod 71.1%

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

      9. sqr-neg 71.1%

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

      10. sqrt-unprod 26.7%

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

      11. add-sqr-sqrt 56.9%

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

      12. sub-neg 56.9%

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

      13. remove-double-neg 56.9%

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

   5. Applied egg-rr 56.9%

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

   6. Taylor expanded in t1 around inf 36.5%

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

      1. div-inv 36.5%

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

      2. frac-2neg 36.5%

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

      3. metadata-eval 36.5%

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

      4. distribute-neg-in 36.5%

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

      5. add-sqr-sqrt 19.4%

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

      6. sqrt-unprod 51.4%

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

      7. sqr-neg 51.4%

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

      8. sqrt-unprod 35.5%

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

      9. add-sqr-sqrt 78.0%

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

      10. sub-neg 78.0%

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

   8. Applied egg-rr 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*l/ 78.2%

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

      3. neg-mul-1 78.2%

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

   10. Simplified 78.2%

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

   if -3e-176 < t1 < 3.09999999999999992e-83

   1. Initial program 83.6%

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

      1. times-frac 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in t1 around 0 83.4%

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

   5. Taylor expanded in t1 around 0 83.7%

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

      1. associate-*r/ 83.7%

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

      2. mul-1-neg 83.7%

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

   7. Simplified 83.7%

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

      1. associate-*l/ 84.2%

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

      2. associate-/l* 81.6%

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

      3. add-sqr-sqrt 34.1%

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

      4. sqrt-unprod 49.3%

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

      5. sqr-neg 49.3%

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

      6. sqrt-unprod 28.9%

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

      7. add-sqr-sqrt 49.0%

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

   9. Applied egg-rr 49.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{-176} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \]

Alternative 6: 65.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.6 \cdot 10^{-171} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.6e-171) (not (<= t1 3.1e-83)))
   (/ (- v) (- t1 u))
   (/ v (* u (/ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.6e-171) || !(t1 <= 3.1e-83)) {
		tmp = -v / (t1 - u);
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4.6d-171)) .or. (.not. (t1 <= 3.1d-83))) then
        tmp = -v / (t1 - u)
    else
        tmp = v / (u * (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.6e-171) || !(t1 <= 3.1e-83)) {
		tmp = -v / (t1 - u);
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.6e-171) or not (t1 <= 3.1e-83):
		tmp = -v / (t1 - u)
	else:
		tmp = v / (u * (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.6e-171) || !(t1 <= 3.1e-83))
		tmp = Float64(Float64(-v) / Float64(t1 - u));
	else
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.6e-171) || ~((t1 <= 3.1e-83)))
		tmp = -v / (t1 - u);
	else
		tmp = v / (u * (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.6e-171], N[Not[LessEqual[t1, 3.1e-83]], $MachinePrecision]], N[((-v) / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.59999999999999956e-171 or 3.09999999999999992e-83 < t1

   1. Initial program 76.1%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. frac-2neg 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. add-sqr-sqrt 54.0%

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

      8. sqrt-unprod 71.1%

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

      9. sqr-neg 71.1%

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

      10. sqrt-unprod 26.7%

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

      11. add-sqr-sqrt 56.9%

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

      12. sub-neg 56.9%

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

      13. remove-double-neg 56.9%

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

   5. Applied egg-rr 56.9%

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

   6. Taylor expanded in t1 around inf 36.5%

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

      1. div-inv 36.5%

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

      2. frac-2neg 36.5%

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

      3. metadata-eval 36.5%

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

      4. distribute-neg-in 36.5%

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

      5. add-sqr-sqrt 19.4%

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

      6. sqrt-unprod 51.4%

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

      7. sqr-neg 51.4%

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

      8. sqrt-unprod 35.5%

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

      9. add-sqr-sqrt 78.0%

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

      10. sub-neg 78.0%

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

   8. Applied egg-rr 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*l/ 78.2%

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

      3. neg-mul-1 78.2%

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

   10. Simplified 78.2%

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

   if -4.59999999999999956e-171 < t1 < 3.09999999999999992e-83

   1. Initial program 83.6%

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

      1. times-frac 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in t1 around 0 83.4%

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

   5. Taylor expanded in t1 around 0 83.7%

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

      1. associate-*r/ 83.7%

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

      2. mul-1-neg 83.7%

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

   7. Simplified 83.7%

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

      1. clear-num 83.2%

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

      2. frac-times 87.0%

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

      3. *-un-lft-identity 87.0%

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

      4. add-sqr-sqrt 34.6%

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

      5. sqrt-unprod 49.7%

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

      6. sqr-neg 49.7%

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

      7. sqrt-unprod 28.9%

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

      8. add-sqr-sqrt 49.1%

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

   9. Applied egg-rr 49.1%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.6 \cdot 10^{-171} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \]

Alternative 7: 65.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -8 \cdot 10^{-177} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -8e-177) (not (<= t1 3.1e-83)))
   (/ (- v) (- t1 u))
   (/ (* v (/ t1 u)) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -8e-177) || !(t1 <= 3.1e-83)) {
		tmp = -v / (t1 - u);
	} else {
		tmp = (v * (t1 / u)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-8d-177)) .or. (.not. (t1 <= 3.1d-83))) then
        tmp = -v / (t1 - u)
    else
        tmp = (v * (t1 / u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -8e-177) || !(t1 <= 3.1e-83)) {
		tmp = -v / (t1 - u);
	} else {
		tmp = (v * (t1 / u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -8e-177) or not (t1 <= 3.1e-83):
		tmp = -v / (t1 - u)
	else:
		tmp = (v * (t1 / u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -8e-177) || !(t1 <= 3.1e-83))
		tmp = Float64(Float64(-v) / Float64(t1 - u));
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -8e-177) || ~((t1 <= 3.1e-83)))
		tmp = -v / (t1 - u);
	else
		tmp = (v * (t1 / u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -8e-177], N[Not[LessEqual[t1, 3.1e-83]], $MachinePrecision]], N[((-v) / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.99999999999999962e-177 or 3.09999999999999992e-83 < t1

   1. Initial program 76.1%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. frac-2neg 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. add-sqr-sqrt 54.0%

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

      8. sqrt-unprod 71.1%

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

      9. sqr-neg 71.1%

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

      10. sqrt-unprod 26.7%

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

      11. add-sqr-sqrt 56.9%

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

      12. sub-neg 56.9%

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

      13. remove-double-neg 56.9%

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

   5. Applied egg-rr 56.9%

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

   6. Taylor expanded in t1 around inf 36.5%

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

      1. div-inv 36.5%

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

      2. frac-2neg 36.5%

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

      3. metadata-eval 36.5%

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

      4. distribute-neg-in 36.5%

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

      5. add-sqr-sqrt 19.4%

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

      6. sqrt-unprod 51.4%

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

      7. sqr-neg 51.4%

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

      8. sqrt-unprod 35.5%

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

      9. add-sqr-sqrt 78.0%

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

      10. sub-neg 78.0%

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

   8. Applied egg-rr 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*l/ 78.2%

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

      3. neg-mul-1 78.2%

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

   10. Simplified 78.2%

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

   if -7.99999999999999962e-177 < t1 < 3.09999999999999992e-83

   1. Initial program 83.6%

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

      1. times-frac 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in t1 around 0 83.4%

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

   5. Taylor expanded in t1 around 0 83.7%

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

      1. associate-*r/ 83.7%

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

      2. mul-1-neg 83.7%

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

   7. Simplified 83.7%

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

      1. associate-*r/ 85.2%

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

      2. *-commutative 85.2%

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

      3. add-sqr-sqrt 34.3%

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

      4. sqrt-unprod 49.7%

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

      5. sqr-neg 49.7%

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

      6. sqrt-unprod 28.9%

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

      7. add-sqr-sqrt 49.1%

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

   9. Applied egg-rr 49.1%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8 \cdot 10^{-177} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{u}\\ \end{array} \]

Alternative 8: 65.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \cdot 10^{-172}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{elif}\;t1 \leq 6.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.4e-172)
   (/ (- v) (- t1 u))
   (if (<= t1 6.4e-77) (/ v (* u (/ u t1))) (/ v (- (* u -2.0) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.4e-172) {
		tmp = -v / (t1 - u);
	} else if (t1 <= 6.4e-77) {
		tmp = v / (u * (u / t1));
	} 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 (t1 <= (-5.4d-172)) then
        tmp = -v / (t1 - u)
    else if (t1 <= 6.4d-77) then
        tmp = v / (u * (u / t1))
    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 (t1 <= -5.4e-172) {
		tmp = -v / (t1 - u);
	} else if (t1 <= 6.4e-77) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.4e-172:
		tmp = -v / (t1 - u)
	elif t1 <= 6.4e-77:
		tmp = v / (u * (u / t1))
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.4e-172)
		tmp = Float64(Float64(-v) / Float64(t1 - u));
	elseif (t1 <= 6.4e-77)
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	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 (t1 <= -5.4e-172)
		tmp = -v / (t1 - u);
	elseif (t1 <= 6.4e-77)
		tmp = v / (u * (u / t1));
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.4e-172], N[((-v) / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6.4e-77], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.40000000000000051e-172

   1. Initial program 75.6%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. frac-2neg 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. add-sqr-sqrt 99.4%

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

      8. sqrt-unprod 83.4%

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

      9. sqr-neg 83.4%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 55.6%

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

      12. sub-neg 55.6%

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

      13. remove-double-neg 55.6%

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

   5. Applied egg-rr 55.6%

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

   6. Taylor expanded in t1 around inf 35.7%

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

      1. div-inv 35.7%

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

      2. frac-2neg 35.7%

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

      3. metadata-eval 35.7%

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

      4. distribute-neg-in 35.7%

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

      5. add-sqr-sqrt 35.7%

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

      6. sqrt-unprod 36.5%

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

      7. sqr-neg 36.5%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 78.1%

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

      10. sub-neg 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*l/ 78.3%

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

      3. neg-mul-1 78.3%

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

   10. Simplified 78.3%

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

   if -5.40000000000000051e-172 < t1 < 6.39999999999999999e-77

   1. Initial program 82.9%

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

      1. times-frac 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in t1 around 0 83.8%

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

   5. Taylor expanded in t1 around 0 84.0%

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

      1. associate-*r/ 84.0%

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

      2. mul-1-neg 84.0%

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

   7. Simplified 84.0%

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

      1. clear-num 83.5%

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

      2. frac-times 86.2%

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

      3. *-un-lft-identity 86.2%

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

      4. add-sqr-sqrt 33.8%

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

      5. sqrt-unprod 48.6%

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

      6. sqr-neg 48.6%

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

      7. sqrt-unprod 28.3%

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

      8. add-sqr-sqrt 48.1%

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

   9. Applied egg-rr 48.1%

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

   if 6.39999999999999999e-77 < t1

   1. Initial program 77.2%

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

      1. associate-/r* 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 92.7%

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

      5. +-commutative 92.7%

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

      6. remove-double-neg 92.7%

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

      7. unsub-neg 92.7%

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

      8. div-sub 92.7%

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

      9. sub-neg 92.7%

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

      10. *-inverses 92.7%

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

      11. metadata-eval 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in t1 around inf 80.0%

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

      1. mul-1-neg 80.0%

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

      2. unsub-neg 80.0%

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

      3. *-commutative 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \cdot 10^{-172}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{elif}\;t1 \leq 6.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 9: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{v}{u + t1}}{-1 - \frac{u}{t1}} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (/ v (+ u t1)) (- -1.0 (/ u t1))))

C

double code(double u, double v, double t1) {
	return (v / (u + t1)) / (-1.0 - (u / t1));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v / (u + t1)) / ((-1.0d0) - (u / t1))
end function

Java

public static double code(double u, double v, double t1) {
	return (v / (u + t1)) / (-1.0 - (u / t1));
}

Python

def code(u, v, t1):
	return (v / (u + t1)) / (-1.0 - (u / t1))

Julia

function code(u, v, t1)
	return Float64(Float64(v / Float64(u + t1)) / Float64(-1.0 - Float64(u / t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v / (u + t1)) / (-1.0 - (u / t1));
end

Wolfram

code[u_, v_, t1_] := N[(N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. associate-/r* 85.6%

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

   2. *-commutative 85.6%

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

   3. associate-/l* 97.5%

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

   4. associate-/l/ 94.7%

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

   5. +-commutative 94.7%

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

   6. remove-double-neg 94.7%

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

   7. unsub-neg 94.7%

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

   8. div-sub 94.7%

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

   9. sub-neg 94.7%

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

   10. *-inverses 94.7%

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

   11. metadata-eval 94.7%

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

3. Simplified 94.7%

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

4. Taylor expanded in v around 0 94.7%

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

   1. associate-/r* 97.1%

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

   2. fma-neg 97.1%

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

   3. metadata-eval 97.1%

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

6. Simplified 97.1%

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

7. Taylor expanded in v around 0 94.7%

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

   1. associate-/r* 97.1%

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

   2. mul-1-neg 97.1%

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

   3. neg-sub0 97.1%

      \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\left(0 - \frac{u}{t1}\right)} - 1} \]

   4. associate--r+ 97.1%

      \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{0 - \left(\frac{u}{t1} + 1\right)}} \]

   5. +-commutative 97.1%

      \[\leadsto \frac{\frac{v}{t1 + u}}{0 - \color{blue}{\left(1 + \frac{u}{t1}\right)}} \]

   6. associate--r+ 97.1%

      \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\left(0 - 1\right) - \frac{u}{t1}}} \]

   7. metadata-eval 97.1%

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

9. Simplified 97.1%

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

10. Final simplification 97.1%

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

Alternative 10: 57.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+85} \lor \neg \left(u \leq 1.35 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.6e+85) (not (<= u 1.35e+103))) (/ v (+ u t1)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.6e+85) || !(u <= 1.35e+103)) {
		tmp = v / (u + t1);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.6d+85)) .or. (.not. (u <= 1.35d+103))) then
        tmp = v / (u + t1)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.6e+85) || !(u <= 1.35e+103)) {
		tmp = v / (u + t1);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.6e+85) or not (u <= 1.35e+103):
		tmp = v / (u + t1)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.6e+85) || !(u <= 1.35e+103))
		tmp = Float64(v / Float64(u + t1));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.6e+85) || ~((u <= 1.35e+103)))
		tmp = v / (u + t1);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.6e+85], N[Not[LessEqual[u, 1.35e+103]], $MachinePrecision]], N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.60000000000000011e85 or 1.34999999999999996e103 < 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 98.2%

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

   3. Simplified 98.2%

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

      1. associate-*r/ 98.2%

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

      2. clear-num 98.2%

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

      3. associate-*l/ 98.2%

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

      4. *-un-lft-identity 98.2%

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

      5. frac-2neg 98.2%

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

      6. distribute-neg-in 98.2%

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

      7. add-sqr-sqrt 46.5%

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

      8. sqrt-unprod 90.2%

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

      9. sqr-neg 90.2%

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

      10. sqrt-unprod 50.4%

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

      11. add-sqr-sqrt 94.8%

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

      12. sub-neg 94.8%

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

      13. remove-double-neg 94.8%

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

   5. Applied egg-rr 94.8%

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

   6. Taylor expanded in t1 around inf 49.8%

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

   if -2.60000000000000011e85 < u < 1.34999999999999996e103

   1. Initial program 75.3%

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

      1. 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 64.6%

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

      1. associate-*r/ 64.6%

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

      2. neg-mul-1 64.6%

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

   6. Simplified 64.6%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+85} \lor \neg \left(u \leq 1.35 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 58.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{+131} \lor \neg \left(u \leq 3.4 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.4e+131) (not (<= u 3.4e+113))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.4e+131) || !(u <= 3.4e+113)) {
		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 <= (-6.4d+131)) .or. (.not. (u <= 3.4d+113))) 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 <= -6.4e+131) || !(u <= 3.4e+113)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.4e+131) or not (u <= 3.4e+113):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.4e+131) || !(u <= 3.4e+113))
		tmp = Float64(v / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.4e+131) || ~((u <= 3.4e+113)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.4e+131], N[Not[LessEqual[u, 3.4e+113]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.4000000000000004e131 or 3.40000000000000019e113 < u

   1. Initial program 82.9%

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

      1. associate-*r/ 97.9%

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

      2. clear-num 97.9%

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

      3. associate-*l/ 97.9%

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

      4. *-un-lft-identity 97.9%

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

      5. frac-2neg 97.9%

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

      6. distribute-neg-in 97.9%

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

      7. add-sqr-sqrt 47.2%

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

      8. sqrt-unprod 89.9%

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

      9. sqr-neg 89.9%

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

      10. sqrt-unprod 50.7%

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

      11. add-sqr-sqrt 96.6%

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

      12. sub-neg 96.6%

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

      13. remove-double-neg 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in t1 around inf 50.8%

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

   7. Taylor expanded in t1 around 0 49.6%

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

   if -6.4000000000000004e131 < u < 3.40000000000000019e113

   1. Initial program 76.7%

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

      1. times-frac 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in t1 around inf 62.5%

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

      1. associate-*r/ 62.5%

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

      2. neg-mul-1 62.5%

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

   6. Simplified 62.5%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{+131} \lor \neg \left(u \leq 3.4 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12: 58.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.5 \cdot 10^{+135}:\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{+119}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.5e+135) (- (/ v u)) (if (<= u 6e+119) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.5e+135) {
		tmp = -(v / u);
	} else if (u <= 6e+119) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.5d+135)) then
        tmp = -(v / u)
    else if (u <= 6d+119) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.5e+135) {
		tmp = -(v / u);
	} else if (u <= 6e+119) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.5e+135:
		tmp = -(v / u)
	elif u <= 6e+119:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.5e+135)
		tmp = Float64(-Float64(v / u));
	elseif (u <= 6e+119)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.5e+135)
		tmp = -(v / u);
	elseif (u <= 6e+119)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.5e+135], (-N[(v / u), $MachinePrecision]), If[LessEqual[u, 6e+119], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.50000000000000015e135

   1. Initial program 76.5%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 93.7%

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

   5. Taylor expanded in t1 around inf 44.1%

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

      1. associate-*r/ 44.1%

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

      2. neg-mul-1 44.1%

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

   7. Simplified 44.1%

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

   if -2.50000000000000015e135 < u < 6.00000000000000002e119

   1. Initial program 76.7%

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

      1. times-frac 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in t1 around inf 62.5%

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

      1. associate-*r/ 62.5%

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

      2. neg-mul-1 62.5%

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

   6. Simplified 62.5%

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

   if 6.00000000000000002e119 < u

   1. Initial program 87.1%

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

      1. times-frac 96.6%

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

   3. Simplified 96.6%

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

      1. associate-*r/ 96.6%

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

      2. clear-num 96.6%

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

      3. associate-*l/ 96.6%

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

      4. *-un-lft-identity 96.6%

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

      5. frac-2neg 96.6%

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

      6. distribute-neg-in 96.6%

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

      7. add-sqr-sqrt 45.4%

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

      8. sqrt-unprod 94.6%

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

      9. sqr-neg 94.6%

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

      10. sqrt-unprod 51.2%

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

      11. add-sqr-sqrt 96.6%

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

      12. sub-neg 96.6%

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

      13. remove-double-neg 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in t1 around inf 53.4%

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

   7. Taylor expanded in t1 around 0 53.4%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.5 \cdot 10^{+135}:\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{+119}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 13: 58.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{+135}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+113}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.6e+135)
   (* (/ v u) -0.5)
   (if (<= u 4.7e+113) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.6e+135) {
		tmp = (v / u) * -0.5;
	} else if (u <= 4.7e+113) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.6d+135)) then
        tmp = (v / u) * (-0.5d0)
    else if (u <= 4.7d+113) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.6e+135) {
		tmp = (v / u) * -0.5;
	} else if (u <= 4.7e+113) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.6e+135:
		tmp = (v / u) * -0.5
	elif u <= 4.7e+113:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.6e+135)
		tmp = Float64(Float64(v / u) * -0.5);
	elseif (u <= 4.7e+113)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.6e+135)
		tmp = (v / u) * -0.5;
	elseif (u <= 4.7e+113)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.6e+135], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[u, 4.7e+113], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.6000000000000002e135

   1. Initial program 76.5%

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

      1. associate-/r* 90.2%

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

      2. *-commutative 90.2%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 84.6%

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

      5. +-commutative 84.6%

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

      6. remove-double-neg 84.6%

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

      7. unsub-neg 84.6%

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

      8. div-sub 84.6%

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

      9. sub-neg 84.6%

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

      10. *-inverses 84.6%

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

      11. metadata-eval 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in t1 around inf 50.3%

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

      1. mul-1-neg 50.3%

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

      2. unsub-neg 50.3%

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

      3. *-commutative 50.3%

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

   6. Simplified 50.3%

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

   7. Taylor expanded in u around inf 44.2%

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

   if -4.6000000000000002e135 < u < 4.6999999999999998e113

   1. Initial program 76.7%

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

      1. times-frac 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in t1 around inf 62.5%

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

      1. associate-*r/ 62.5%

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

      2. neg-mul-1 62.5%

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

   6. Simplified 62.5%

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

   if 4.6999999999999998e113 < u

   1. Initial program 87.1%

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

      1. times-frac 96.6%

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

   3. Simplified 96.6%

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

      1. associate-*r/ 96.6%

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

      2. clear-num 96.6%

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

      3. associate-*l/ 96.6%

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

      4. *-un-lft-identity 96.6%

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

      5. frac-2neg 96.6%

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

      6. distribute-neg-in 96.6%

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

      7. add-sqr-sqrt 45.4%

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

      8. sqrt-unprod 94.6%

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

      9. sqr-neg 94.6%

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

      10. sqrt-unprod 51.2%

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

      11. add-sqr-sqrt 96.6%

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

      12. sub-neg 96.6%

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

      13. remove-double-neg 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in t1 around inf 53.4%

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

   7. Taylor expanded in t1 around 0 53.4%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{+135}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+113}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 14: 23.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.95 \cdot 10^{+104} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.95e+104) (not (<= t1 8.5e+111))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.95e+104) || !(t1 <= 8.5e+111)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.95d+104)) .or. (.not. (t1 <= 8.5d+111))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.95e+104) || !(t1 <= 8.5e+111)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.95e+104) or not (t1 <= 8.5e+111):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.95e+104) || !(t1 <= 8.5e+111))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.95e+104) || ~((t1 <= 8.5e+111)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.95e+104], N[Not[LessEqual[t1, 8.5e+111]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.95000000000000008e104 or 8.49999999999999983e111 < t1

   1. Initial program 61.4%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 98.0%

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

      3. frac-2neg 98.0%

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

      4. frac-times 71.5%

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

      5. *-un-lft-identity 71.5%

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

      6. remove-double-neg 71.5%

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

      7. distribute-neg-in 71.5%

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

      8. add-sqr-sqrt 35.3%

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

      9. sqrt-unprod 58.6%

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

      10. sqr-neg 58.6%

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

      11. sqrt-unprod 31.0%

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

      12. add-sqr-sqrt 57.1%

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

      13. sub-neg 57.1%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. associate-*r/ 54.7%

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

   7. Simplified 54.7%

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

   8. Taylor expanded in t1 around inf 51.4%

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

   if -1.95000000000000008e104 < t1 < 8.49999999999999983e111

   1. Initial program 85.6%

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

      1. times-frac 96.1%

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

   3. Simplified 96.1%

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

      1. associate-*r/ 96.8%

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

      2. clear-num 96.5%

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

      3. associate-*l/ 96.5%

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

      4. *-un-lft-identity 96.5%

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

      5. frac-2neg 96.5%

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

      6. distribute-neg-in 96.5%

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

      7. add-sqr-sqrt 48.2%

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

      8. sqrt-unprod 80.9%

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

      9. sqr-neg 80.9%

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

      10. sqrt-unprod 35.1%

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

      11. add-sqr-sqrt 66.8%

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

      12. sub-neg 66.8%

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

      13. remove-double-neg 66.8%

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

   5. Applied egg-rr 66.8%

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

   6. Taylor expanded in t1 around inf 20.3%

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

   7. Taylor expanded in t1 around 0 19.6%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.95 \cdot 10^{+104} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 15: 61.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. times-frac 97.2%

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

3. Simplified 97.2%

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

   1. associate-*r/ 97.7%

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

   2. clear-num 97.5%

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

   3. associate-*l/ 97.5%

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

   4. *-un-lft-identity 97.5%

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

   5. frac-2neg 97.5%

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

   6. distribute-neg-in 97.5%

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

   7. add-sqr-sqrt 48.9%

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

   8. sqrt-unprod 75.4%

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

   9. sqr-neg 75.4%

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

   10. sqrt-unprod 34.5%

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

   11. add-sqr-sqrt 65.8%

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

   12. sub-neg 65.8%

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

   13. remove-double-neg 65.8%

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

5. Applied egg-rr 65.8%

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

   1. frac-2neg 65.8%

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

   2. distribute-frac-neg 65.8%

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

   3. add-sqr-sqrt 31.3%

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

   4. sqrt-unprod 45.1%

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

   5. sqr-neg 45.1%

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

   6. sqrt-unprod 35.6%

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

   7. add-sqr-sqrt 74.3%

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

   8. neg-sub0 74.3%

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

   9. div-sub 74.3%

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

   10. *-inverses 74.3%

       \[\leadsto \frac{\frac{v}{0 - \left(\color{blue}{1} - \frac{u}{t1}\right)}}{t1 + u} \]

   11. sub-neg 74.3%

       \[\leadsto \frac{\frac{v}{0 - \color{blue}{\left(1 + \left(-\frac{u}{t1}\right)\right)}}}{t1 + u} \]

   12. add-sqr-sqrt 35.7%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \left(-\frac{u}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}\right)\right)}}{t1 + u} \]

   13. sqrt-unprod 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \left(-\frac{u}{\color{blue}{\sqrt{t1 \cdot t1}}}\right)\right)}}{t1 + u} \]

   14. sqr-neg 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \left(-\frac{u}{\sqrt{\color{blue}{\left(-t1\right) \cdot \left(-t1\right)}}}\right)\right)}}{t1 + u} \]

   15. sqrt-unprod 49.1%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \left(-\frac{u}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}\right)\right)}}{t1 + u} \]

   16. add-sqr-sqrt 97.5%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \left(-\frac{u}{\color{blue}{-t1}}\right)\right)}}{t1 + u} \]

   17. distribute-frac-neg 97.5%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \color{blue}{\frac{-u}{-t1}}\right)}}{t1 + u} \]

   18. frac-2neg 97.5%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \color{blue}{\frac{u}{t1}}\right)}}{t1 + u} \]

7. Applied egg-rr 97.5%

   \[\leadsto \frac{\frac{v}{\color{blue}{0 - \left(1 + \frac{u}{t1}\right)}}}{t1 + u} \]
8. Step-by-step derivation

   1. associate--r+ 97.5%

      \[\leadsto \frac{\frac{v}{\color{blue}{\left(0 - 1\right) - \frac{u}{t1}}}}{t1 + u} \]

   2. metadata-eval 97.5%

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

9. Simplified 97.5%

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

10. Taylor expanded in u around 0 60.2%

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

    1. neg-mul-1 60.2%

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

12. Simplified 60.2%

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

13. Final simplification 60.2%

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

Alternative 16: 14.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. times-frac 97.2%

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

3. Simplified 97.2%

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

   1. *-commutative 97.2%

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

   2. clear-num 96.4%

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

   3. frac-2neg 96.4%

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

   4. frac-times 84.0%

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

   5. *-un-lft-identity 84.0%

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

   6. remove-double-neg 84.0%

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

   7. distribute-neg-in 84.0%

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

   8. add-sqr-sqrt 41.7%

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

   9. sqrt-unprod 71.2%

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

   10. sqr-neg 71.2%

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

   11. sqrt-unprod 32.1%

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

   12. add-sqr-sqrt 62.3%

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

   13. sub-neg 62.3%

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

5. Applied egg-rr 62.3%

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

   1. *-commutative 62.3%

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

   2. associate-*r/ 59.0%

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

7. Simplified 59.0%

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

8. Taylor expanded in t1 around inf 18.7%

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

9. Final simplification 18.7%

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

Reproduce

herbie shell --seed 2023334 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))