Rosa's DopplerBench

Percentage Accurate: 72.3% → 98.0%

Time: 8.7s

Alternatives: 12

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: 72.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. times-frac 97.3%

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

3. Simplified 97.3%

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

4. Final simplification 97.3%

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

Alternative 2: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-21} \lor \neg \left(t1 \leq 9.8 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.8e-21) (not (<= t1 9.8e+56)))
   (/ v (- (* u -2.0) t1))
   (* (/ v u) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e-21) || !(t1 <= 9.8e+56)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.8d-21)) .or. (.not. (t1 <= 9.8d+56))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (v / u) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e-21) || !(t1 <= 9.8e+56)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.8e-21) or not (t1 <= 9.8e+56):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.8e-21) || !(t1 <= 9.8e+56))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.8e-21) || ~((t1 <= 9.8e+56)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.8e-21], N[Not[LessEqual[t1, 9.8e+56]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.80000000000000004e-21 or 9.8000000000000005e56 < t1

   1. Initial program 58.2%

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

      1. *-commutative 58.2%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 98.3%

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

      1. associate-*r/ 98.3%

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

      2. neg-mul-1 98.3%

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

      3. +-commutative 98.3%

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

   6. Simplified 98.3%

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

      1. frac-2neg 98.3%

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

      2. div-inv 98.0%

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

      3. add-sqr-sqrt 60.3%

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

      4. sqrt-unprod 58.1%

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

      5. sqr-neg 58.1%

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

      6. sqrt-unprod 12.0%

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

      7. add-sqr-sqrt 32.5%

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

      8. add-sqr-sqrt 20.6%

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

      9. sqrt-unprod 48.2%

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

      10. sqr-neg 48.2%

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

      11. sqrt-unprod 37.5%

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

      12. add-sqr-sqrt 98.0%

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

   8. Applied egg-rr 98.0%

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

      1. associate-*r/ 98.3%

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

      2. *-rgt-identity 98.3%

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

      3. distribute-rgt-neg-in 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. metadata-eval 98.3%

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

      7. sub-neg 98.3%

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

   10. Simplified 98.3%

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

   11. Taylor expanded in t1 around inf 86.3%

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

       1. +-commutative 86.3%

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

       2. mul-1-neg 86.3%

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

       3. unsub-neg 86.3%

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

       4. *-commutative 86.3%

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

   13. Simplified 86.3%

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

   if -2.80000000000000004e-21 < t1 < 9.8000000000000005e56

   1. Initial program 81.1%

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

   2. Taylor expanded in t1 around 0 72.2%

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

      1. unpow2 72.2%

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

   4. Simplified 72.2%

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

      1. expm1-log1p-u 63.0%

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

      2. expm1-udef 50.8%

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

      3. div-inv 50.4%

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

      4. associate-*l* 49.3%

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

      5. add-sqr-sqrt 23.0%

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

      6. sqrt-unprod 45.3%

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

      7. sqr-neg 45.3%

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

      8. sqrt-unprod 22.4%

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

      9. add-sqr-sqrt 44.6%

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

      10. pow2 44.6%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot \frac{1}{\color{blue}{{u}^{2}}}\right)\right)} - 1 \]

      11. pow-flip 44.6%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot \color{blue}{{u}^{\left(-2\right)}}\right)\right)} - 1 \]

      12. metadata-eval 44.6%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{\color{blue}{-2}}\right)\right)} - 1 \]

   6. Applied egg-rr 44.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{-2}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 44.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{-2}\right)\right)\right)} \]

      2. expm1-log1p 44.3%

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

      3. associate-*r* 44.4%

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

      4. *-commutative 44.4%

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

      5. metadata-eval 44.4%

         \[\leadsto \left(v \cdot t1\right) \cdot {u}^{\color{blue}{\left(2 \cdot -1\right)}} \]

      6. pow-sqr 44.4%

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

      7. unpow-1 44.4%

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

      8. unpow-1 44.4%

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

      9. associate-*r* 44.6%

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

      10. associate-*r/ 44.6%

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

      11. associate-*l/ 44.6%

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

      12. *-rgt-identity 44.6%

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

      13. associate-*r/ 44.6%

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

      14. *-rgt-identity 44.6%

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

      15. associate-*r/ 44.4%

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

      16. associate-*r/ 44.5%

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

      17. associate-/r* 44.4%

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

   8. Simplified 44.4%

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

      1. add-sqr-sqrt 18.6%

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

      2. sqrt-unprod 50.4%

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

      3. sqr-neg 50.4%

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

      4. sqrt-unprod 40.7%

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

      5. add-sqr-sqrt 72.8%

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

      6. distribute-lft-neg-out 72.8%

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

      7. associate-*r/ 72.2%

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

      8. times-frac 80.0%

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

   10. Applied egg-rr 80.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-21} \lor \neg \left(t1 \leq 9.8 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 3: 66.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{-14} \lor \neg \left(u \leq 1.2 \cdot 10^{+116}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.5e-14) (not (<= u 1.2e+116)))
   (* v (/ t1 (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.5e-14) || !(u <= 1.2e+116)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.5d-14)) .or. (.not. (u <= 1.2d+116))) then
        tmp = v * (t1 / (u * u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.5e-14) || !(u <= 1.2e+116)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.5e-14) or not (u <= 1.2e+116):
		tmp = v * (t1 / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.5e-14) || !(u <= 1.2e+116))
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.5e-14) || ~((u <= 1.2e+116)))
		tmp = v * (t1 / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.5e-14], N[Not[LessEqual[u, 1.2e+116]], $MachinePrecision]], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.4999999999999999e-14 or 1.2e116 < u

   1. Initial program 74.6%

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

   2. Taylor expanded in t1 around 0 71.3%

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

      1. unpow2 71.3%

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

   4. Simplified 71.3%

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

      1. expm1-log1p-u 70.4%

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

      2. expm1-udef 63.6%

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

      3. div-inv 63.6%

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

      4. associate-*l* 64.1%

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

      5. add-sqr-sqrt 31.5%

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

      6. sqrt-unprod 58.6%

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

      7. sqr-neg 58.6%

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

      8. sqrt-unprod 32.4%

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

      9. add-sqr-sqrt 63.8%

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

      10. pow2 63.8%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot \frac{1}{\color{blue}{{u}^{2}}}\right)\right)} - 1 \]

      11. pow-flip 63.8%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot \color{blue}{{u}^{\left(-2\right)}}\right)\right)} - 1 \]

      12. metadata-eval 63.8%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{\color{blue}{-2}}\right)\right)} - 1 \]

   6. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{-2}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 63.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{-2}\right)\right)\right)} \]

      2. expm1-log1p 63.6%

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

      3. associate-*r* 63.2%

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

      4. *-commutative 63.2%

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

      5. metadata-eval 63.2%

         \[\leadsto \left(v \cdot t1\right) \cdot {u}^{\color{blue}{\left(2 \cdot -1\right)}} \]

      6. pow-sqr 63.2%

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

      7. unpow-1 63.2%

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

      8. unpow-1 63.2%

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

      9. associate-*r* 63.0%

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

      10. associate-*r/ 63.0%

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

      11. associate-*l/ 63.0%

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

      12. *-rgt-identity 63.0%

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

      13. associate-*r/ 63.0%

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

      14. *-rgt-identity 63.0%

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

      15. associate-*r/ 63.1%

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

      16. associate-*r/ 63.3%

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

      17. associate-/r* 63.7%

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

   8. Simplified 63.7%

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

   if -1.4999999999999999e-14 < u < 1.2e116

   1. Initial program 68.1%

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

      1. times-frac 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in t1 around inf 68.7%

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

      1. associate-*r/ 68.7%

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

      2. neg-mul-1 68.7%

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

   6. Simplified 68.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{-14} \lor \neg \left(u \leq 1.2 \cdot 10^{+116}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 4: 66.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{-14}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.4e-14)
   (* v (/ t1 (* u u)))
   (if (<= u 6.2e+114) (/ (- v) t1) (/ v (/ (* u u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e-14) {
		tmp = v * (t1 / (u * u));
	} else if (u <= 6.2e+114) {
		tmp = -v / t1;
	} else {
		tmp = v / ((u * u) / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.4d-14)) then
        tmp = v * (t1 / (u * u))
    else if (u <= 6.2d+114) then
        tmp = -v / t1
    else
        tmp = v / ((u * u) / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e-14) {
		tmp = v * (t1 / (u * u));
	} else if (u <= 6.2e+114) {
		tmp = -v / t1;
	} else {
		tmp = v / ((u * u) / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.4e-14:
		tmp = v * (t1 / (u * u))
	elif u <= 6.2e+114:
		tmp = -v / t1
	else:
		tmp = v / ((u * u) / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.4e-14)
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	elseif (u <= 6.2e+114)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / Float64(Float64(u * u) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.4e-14)
		tmp = v * (t1 / (u * u));
	elseif (u <= 6.2e+114)
		tmp = -v / t1;
	else
		tmp = v / ((u * u) / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.4e-14], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 6.2e+114], N[((-v) / t1), $MachinePrecision], N[(v / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.4e-14

   1. Initial program 75.7%

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

   2. Taylor expanded in t1 around 0 69.3%

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

      1. unpow2 69.3%

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

   4. Simplified 69.3%

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

      1. expm1-log1p-u 67.8%

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

      2. expm1-udef 58.2%

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

      3. div-inv 58.2%

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

      4. associate-*l* 58.5%

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

      5. add-sqr-sqrt 35.0%

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

      6. sqrt-unprod 53.6%

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

      7. sqr-neg 53.6%

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

      8. sqrt-unprod 22.9%

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

      9. add-sqr-sqrt 57.9%

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

      10. pow2 57.9%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot \frac{1}{\color{blue}{{u}^{2}}}\right)\right)} - 1 \]

      11. pow-flip 57.9%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot \color{blue}{{u}^{\left(-2\right)}}\right)\right)} - 1 \]

      12. metadata-eval 57.9%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{\color{blue}{-2}}\right)\right)} - 1 \]

   6. Applied egg-rr 57.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{-2}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 57.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{-2}\right)\right)\right)} \]

      2. expm1-log1p 57.6%

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

      3. associate-*r* 57.3%

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

      4. *-commutative 57.3%

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

      5. metadata-eval 57.3%

         \[\leadsto \left(v \cdot t1\right) \cdot {u}^{\color{blue}{\left(2 \cdot -1\right)}} \]

      6. pow-sqr 57.3%

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

      7. unpow-1 57.3%

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

      8. unpow-1 57.3%

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

      9. associate-*r* 56.9%

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

      10. associate-*r/ 56.9%

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

      11. associate-*l/ 56.9%

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

      12. *-rgt-identity 56.9%

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

      13. associate-*r/ 56.9%

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

      14. *-rgt-identity 56.9%

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

      15. associate-*r/ 57.0%

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

      16. associate-*r/ 57.2%

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

      17. associate-/r* 57.8%

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

   8. Simplified 57.8%

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

   if -1.4e-14 < u < 6.2000000000000001e114

   1. Initial program 68.1%

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

      1. times-frac 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in t1 around inf 68.7%

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

      1. associate-*r/ 68.7%

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

      2. neg-mul-1 68.7%

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

   6. Simplified 68.7%

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

   if 6.2000000000000001e114 < u

   1. Initial program 73.4%

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

   2. Taylor expanded in t1 around 0 73.5%

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

      1. unpow2 73.5%

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

   4. Simplified 73.5%

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

      1. expm1-log1p-u 73.4%

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

      2. expm1-udef 69.8%

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

      3. div-inv 69.8%

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

      4. associate-*l* 70.5%

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

      5. add-sqr-sqrt 27.4%

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

      6. sqrt-unprod 64.2%

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

      7. sqr-neg 64.2%

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

      8. sqrt-unprod 43.1%

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

      9. add-sqr-sqrt 70.6%

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

      10. pow2 70.6%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot \frac{1}{\color{blue}{{u}^{2}}}\right)\right)} - 1 \]

      11. pow-flip 70.5%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot \color{blue}{{u}^{\left(-2\right)}}\right)\right)} - 1 \]

      12. metadata-eval 70.5%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{\color{blue}{-2}}\right)\right)} - 1 \]

   6. Applied egg-rr 70.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{-2}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 70.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{-2}\right)\right)\right)} \]

      2. expm1-log1p 70.5%

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

      3. associate-*r* 69.8%

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

      4. *-commutative 69.8%

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

      5. metadata-eval 69.8%

         \[\leadsto \left(v \cdot t1\right) \cdot {u}^{\color{blue}{\left(2 \cdot -1\right)}} \]

      6. pow-sqr 69.8%

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

      7. unpow-1 69.8%

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

      8. unpow-1 69.8%

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

      9. associate-*r* 69.8%

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

      10. associate-*r/ 69.8%

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

      11. associate-*l/ 69.8%

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

      12. *-rgt-identity 69.8%

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

      13. associate-*r/ 69.8%

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

      14. *-rgt-identity 69.8%

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

      15. associate-*r/ 70.1%

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

      16. associate-*r/ 70.1%

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

      17. associate-/r* 70.5%

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

   8. Simplified 70.5%

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

   9. Taylor expanded in v around 0 69.8%

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

       1. unpow2 69.8%

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

       2. *-commutative 69.8%

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

       3. associate-/l* 70.5%

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

   11. Simplified 70.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{-14}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \end{array} \]

Alternative 5: 95.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)} \end{array} \]

FPCore

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

C

double code(double u, double v, double t1) {
	return v / ((t1 + u) * (-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 / ((t1 + u) * ((-1.0d0) - (u / t1)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. *-commutative 70.9%

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

   2. times-frac 97.3%

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

   3. neg-mul-1 97.3%

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

   4. associate-/l* 97.3%

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

   5. associate-*r/ 97.3%

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

   6. associate-/l* 97.3%

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

   7. associate-/l/ 97.3%

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

   8. neg-mul-1 97.3%

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

   9. *-lft-identity 97.3%

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

   10. metadata-eval 97.3%

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

   11. times-frac 97.3%

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

   12. neg-mul-1 97.3%

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

   13. remove-double-neg 97.3%

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

   14. neg-mul-1 97.3%

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

   15. sub0-neg 97.3%

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

   16. associate--r+ 97.3%

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

   17. neg-sub0 97.3%

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

   18. div-sub 97.3%

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

   19. distribute-frac-neg 97.3%

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

   20. *-inverses 97.3%

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

   21. metadata-eval 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in v around 0 95.5%

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

   1. associate-*r/ 95.5%

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

   2. neg-mul-1 95.5%

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

   3. +-commutative 95.5%

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

6. Simplified 95.5%

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

   1. frac-2neg 95.5%

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

   2. div-inv 95.3%

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

   3. add-sqr-sqrt 53.8%

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

   4. sqrt-unprod 56.5%

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

   5. sqr-neg 56.5%

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

   6. sqrt-unprod 15.7%

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

   7. add-sqr-sqrt 39.2%

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

   8. add-sqr-sqrt 23.5%

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

   9. sqrt-unprod 53.1%

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

   10. sqr-neg 53.1%

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

   11. sqrt-unprod 41.3%

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

   12. add-sqr-sqrt 95.3%

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

8. Applied egg-rr 95.3%

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

   1. associate-*r/ 95.5%

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

   2. *-rgt-identity 95.5%

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

   3. distribute-rgt-neg-in 95.5%

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

   4. +-commutative 95.5%

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

   5. distribute-neg-in 95.5%

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

   6. metadata-eval 95.5%

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

   7. sub-neg 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

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

Alternative 6: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return (v / (t1 + u)) / (-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 / (t1 + u)) / ((-1.0d0) - (u / t1))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. *-commutative 70.9%

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

   2. times-frac 97.3%

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

   3. neg-mul-1 97.3%

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

   4. associate-/l* 97.3%

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

   5. associate-*r/ 97.3%

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

   6. associate-/l* 97.3%

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

   7. associate-/l/ 97.3%

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

   8. neg-mul-1 97.3%

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

   9. *-lft-identity 97.3%

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

   10. metadata-eval 97.3%

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

   11. times-frac 97.3%

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

   12. neg-mul-1 97.3%

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

   13. remove-double-neg 97.3%

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

   14. neg-mul-1 97.3%

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

   15. sub0-neg 97.3%

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

   16. associate--r+ 97.3%

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

   17. neg-sub0 97.3%

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

   18. div-sub 97.3%

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

   19. distribute-frac-neg 97.3%

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

   20. *-inverses 97.3%

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

   21. metadata-eval 97.3%

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

3. Simplified 97.3%

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

4. Final simplification 97.3%

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

Alternative 7: 58.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.9 \cdot 10^{+192}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.9e+192)
   (/ (- v) u)
   (if (<= u 9.5e+99) (/ (- v) t1) (/ 1.0 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.9e+192) {
		tmp = -v / u;
	} else if (u <= 9.5e+99) {
		tmp = -v / t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.9d+192)) then
        tmp = -v / u
    else if (u <= 9.5d+99) then
        tmp = -v / t1
    else
        tmp = 1.0d0 / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.9e+192) {
		tmp = -v / u;
	} else if (u <= 9.5e+99) {
		tmp = -v / t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.9e+192:
		tmp = -v / u
	elif u <= 9.5e+99:
		tmp = -v / t1
	else:
		tmp = 1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.9e+192)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 9.5e+99)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(1.0 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.9e+192)
		tmp = -v / u;
	elseif (u <= 9.5e+99)
		tmp = -v / t1;
	else
		tmp = 1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.9e+192], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 9.5e+99], N[((-v) / t1), $MachinePrecision], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.90000000000000047e192

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 56.8%

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

   5. Taylor expanded in t1 around 0 40.6%

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

      1. associate-*r/ 40.6%

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

      2. neg-mul-1 40.6%

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

   7. Simplified 40.6%

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

   if -4.90000000000000047e192 < u < 9.49999999999999908e99

   1. Initial program 68.8%

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

      1. times-frac 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in t1 around inf 62.4%

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

      1. associate-*r/ 62.4%

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

      2. neg-mul-1 62.4%

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

   6. Simplified 62.4%

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

   if 9.49999999999999908e99 < u

   1. Initial program 74.9%

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

      1. associate-/l* 75.6%

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

      2. associate-/l* 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in t1 around inf 40.9%

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

      1. clear-num 44.1%

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

      2. inv-pow 44.1%

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

      3. div-inv 44.1%

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

      4. clear-num 44.1%

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

      5. add-sqr-sqrt 15.5%

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

      6. sqrt-unprod 37.4%

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

      7. sqr-neg 37.4%

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

      8. sqrt-unprod 24.7%

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

      9. add-sqr-sqrt 40.5%

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

   6. Applied egg-rr 40.5%

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

      1. unpow-1 40.5%

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

      2. associate-/l* 40.5%

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

      3. associate-/r/ 40.7%

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

      4. *-inverses 40.7%

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

      5. *-lft-identity 40.7%

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

   8. Simplified 40.7%

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

   9. Taylor expanded in t1 around 0 40.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{u}{v}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.9 \cdot 10^{+192}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]

Alternative 8: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+192} \lor \neg \left(u \leq 6 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.6e+192) (not (<= u 6e+116))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.6e+192) || !(u <= 6e+116)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.6d+192)) .or. (.not. (u <= 6d+116))) then
        tmp = -v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.6e+192) || !(u <= 6e+116)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.6e+192) or not (u <= 6e+116):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.6e+192) || !(u <= 6e+116))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.6e+192) || ~((u <= 6e+116)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.6e+192], N[Not[LessEqual[u, 6e+116]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.60000000000000003e192 or 5.9999999999999997e116 < u

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.8%

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

      6. associate-/l* 99.8%

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

      7. associate-/l/ 99.8%

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

      8. neg-mul-1 99.8%

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

      9. *-lft-identity 99.8%

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

      10. metadata-eval 99.8%

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

      11. times-frac 99.8%

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

      12. neg-mul-1 99.8%

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

      13. remove-double-neg 99.8%

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

      14. neg-mul-1 99.8%

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

      15. sub0-neg 99.8%

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

      16. associate--r+ 99.8%

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

      17. neg-sub0 99.8%

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

      18. div-sub 99.8%

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

      19. distribute-frac-neg 99.8%

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

      20. *-inverses 99.8%

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

      21. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 56.7%

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

   5. Taylor expanded in t1 around 0 38.9%

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

      1. associate-*r/ 38.9%

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

      2. neg-mul-1 38.9%

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

   7. Simplified 38.9%

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

   if -2.60000000000000003e192 < u < 5.9999999999999997e116

   1. Initial program 69.3%

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

      1. times-frac 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in t1 around inf 62.0%

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

      1. associate-*r/ 62.0%

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

      2. neg-mul-1 62.0%

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

   6. Simplified 62.0%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+192} \lor \neg \left(u \leq 6 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 9: 58.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.5 \cdot 10^{+192}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.1 \cdot 10^{+100}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7.5e+192) (/ v u) (if (<= u 1.1e+100) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.5e+192) {
		tmp = v / u;
	} else if (u <= 1.1e+100) {
		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 <= (-7.5d+192)) then
        tmp = v / u
    else if (u <= 1.1d+100) 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 <= -7.5e+192) {
		tmp = v / u;
	} else if (u <= 1.1e+100) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7.5e+192:
		tmp = v / u
	elif u <= 1.1e+100:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7.5e+192)
		tmp = Float64(v / u);
	elseif (u <= 1.1e+100)
		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 <= -7.5e+192)
		tmp = v / u;
	elseif (u <= 1.1e+100)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7.5e+192], N[(v / u), $MachinePrecision], If[LessEqual[u, 1.1e+100], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -7.5e192 or 1.1e100 < u

   1. Initial program 75.8%

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

      1. associate-/l* 76.5%

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

      2. associate-/l* 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in t1 around inf 40.6%

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

      1. clear-num 42.9%

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

      2. inv-pow 42.9%

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

      3. div-inv 42.9%

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

      4. clear-num 42.9%

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

      5. add-sqr-sqrt 18.0%

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

      6. sqrt-unprod 40.2%

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

      7. sqr-neg 40.2%

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

      8. sqrt-unprod 22.1%

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

      9. add-sqr-sqrt 40.4%

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

   6. Applied egg-rr 40.4%

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

      1. unpow-1 40.4%

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

      2. associate-/l* 40.3%

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

      3. associate-/r/ 40.6%

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

      4. *-inverses 40.6%

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

      5. *-lft-identity 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in t1 around 0 38.2%

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

   if -7.5e192 < u < 1.1e100

   1. Initial program 68.8%

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

      1. times-frac 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in t1 around inf 62.4%

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

      1. associate-*r/ 62.4%

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

      2. neg-mul-1 62.4%

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

   6. Simplified 62.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.5 \cdot 10^{+192}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.1 \cdot 10^{+100}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 10: 22.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 2.85 \cdot 10^{+215}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.2e+158) (/ v t1) (if (<= t1 2.85e+215) (/ v u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+158) {
		tmp = v / t1;
	} else if (t1 <= 2.85e+215) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.2d+158)) then
        tmp = v / t1
    else if (t1 <= 2.85d+215) then
        tmp = v / u
    else
        tmp = v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+158) {
		tmp = v / t1;
	} else if (t1 <= 2.85e+215) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.2e+158:
		tmp = v / t1
	elif t1 <= 2.85e+215:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.2e+158)
		tmp = Float64(v / t1);
	elseif (t1 <= 2.85e+215)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.2e+158)
		tmp = v / t1;
	elseif (t1 <= 2.85e+215)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.2e+158], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 2.85e+215], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.20000000000000004e158 or 2.85e215 < t1

   1. Initial program 45.8%

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

      1. associate-/l* 47.3%

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

      2. associate-/l* 62.6%

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

   3. Simplified 62.6%

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

      1. frac-2neg 62.6%

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

      2. div-inv 62.6%

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

      3. distribute-neg-in 62.6%

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

      4. add-sqr-sqrt 40.5%

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

      5. sqrt-unprod 47.3%

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

      6. sqr-neg 47.3%

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

      7. sqrt-unprod 13.5%

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

      8. add-sqr-sqrt 47.1%

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

      9. sub-neg 47.1%

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

      10. distribute-neg-frac 47.1%

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

   5. Applied egg-rr 47.1%

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

      1. associate-/r/ 47.1%

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

   7. Simplified 47.1%

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

   8. Taylor expanded in t1 around inf 46.4%

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

   if -1.20000000000000004e158 < t1 < 2.85e215

   1. Initial program 78.1%

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

      1. associate-/l* 78.4%

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

      2. associate-/l* 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in t1 around inf 41.2%

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

      1. clear-num 42.4%

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

      2. inv-pow 42.4%

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

      3. div-inv 42.4%

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

      4. clear-num 42.3%

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

      5. add-sqr-sqrt 19.5%

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

      6. sqrt-unprod 37.2%

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

      7. sqr-neg 37.2%

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

      8. sqrt-unprod 11.2%

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

      9. add-sqr-sqrt 19.8%

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

   6. Applied egg-rr 19.8%

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

      1. unpow-1 19.8%

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

      2. associate-/l* 19.8%

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

      3. associate-/r/ 20.0%

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

      4. *-inverses 20.0%

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

      5. *-lft-identity 20.0%

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

   8. Simplified 20.0%

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

   9. Taylor expanded in t1 around 0 20.2%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 2.85 \cdot 10^{+215}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 11: 62.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. associate-/l* 71.5%

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

   2. associate-/l* 82.5%

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

3. Simplified 82.5%

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

4. Taylor expanded in t1 around inf 45.9%

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

5. Taylor expanded in v around 0 57.2%

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

   1. associate-*r/ 57.2%

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

   2. neg-mul-1 57.2%

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

7. Simplified 57.2%

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

8. Final simplification 57.2%

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

Alternative 12: 14.5% 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 70.9%

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

   1. associate-/l* 71.5%

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

   2. associate-/l* 82.5%

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

3. Simplified 82.5%

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

   1. frac-2neg 82.5%

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

   2. div-inv 82.5%

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

   3. distribute-neg-in 82.5%

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

   4. add-sqr-sqrt 38.2%

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

   5. sqrt-unprod 68.2%

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

   6. sqr-neg 68.2%

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

   7. sqrt-unprod 32.3%

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

   8. add-sqr-sqrt 59.9%

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

   9. sub-neg 59.9%

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

   10. distribute-neg-frac 59.9%

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

5. Applied egg-rr 59.9%

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

   1. associate-/r/ 59.9%

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

7. Simplified 59.9%

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

8. Taylor expanded in t1 around inf 14.0%

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

9. Final simplification 14.0%

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

Reproduce

herbie shell --seed 2023182 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))