Rosa's DopplerBench

Percentage Accurate: 72.1% → 98.2%

Time: 13.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.1% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. associate-/r* 79.1%

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

   2. associate-/l* 97.0%

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

3. Simplified 97.0%

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

   1. div-inv 96.8%

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

   2. clear-num 97.0%

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

   3. add-sqr-sqrt 45.5%

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

   4. sqrt-unprod 39.5%

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

   5. sqr-neg 39.5%

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

   6. sqrt-unprod 17.3%

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

   7. add-sqr-sqrt 34.7%

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

   8. distribute-lft-neg-in 34.7%

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

   9. distribute-rgt-neg-in 34.7%

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

   10. add-sqr-sqrt 17.3%

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

   11. sqrt-unprod 39.5%

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

   12. sqr-neg 39.5%

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

   13. sqrt-unprod 45.5%

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

   14. add-sqr-sqrt 97.0%

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

5. Applied egg-rr 97.0%

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

6. Final simplification 97.0%

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

Alternative 2: 90.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -3.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -4.5 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.15 \cdot 10^{-202}:\\ \;\;\;\;v \cdot \left(\frac{-1}{u} \cdot \frac{t1}{u}\right)\\ \mathbf{elif}\;t1 \leq 1.05 \cdot 10^{+119}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -3.4e+123)
     (/ (- v) (+ t1 u))
     (if (<= t1 -4.5e-148)
       t_1
       (if (<= t1 1.15e-202)
         (* v (* (/ -1.0 u) (/ t1 u)))
         (if (<= t1 1.05e+119) t_1 (/ v (- u t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -3.4e+123) {
		tmp = -v / (t1 + u);
	} else if (t1 <= -4.5e-148) {
		tmp = t_1;
	} else if (t1 <= 1.15e-202) {
		tmp = v * ((-1.0 / u) * (t1 / u));
	} else if (t1 <= 1.05e+119) {
		tmp = t_1;
	} else {
		tmp = v / (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) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-3.4d+123)) then
        tmp = -v / (t1 + u)
    else if (t1 <= (-4.5d-148)) then
        tmp = t_1
    else if (t1 <= 1.15d-202) then
        tmp = v * (((-1.0d0) / u) * (t1 / u))
    else if (t1 <= 1.05d+119) then
        tmp = t_1
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -3.4e+123) {
		tmp = -v / (t1 + u);
	} else if (t1 <= -4.5e-148) {
		tmp = t_1;
	} else if (t1 <= 1.15e-202) {
		tmp = v * ((-1.0 / u) * (t1 / u));
	} else if (t1 <= 1.05e+119) {
		tmp = t_1;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -3.4e+123:
		tmp = -v / (t1 + u)
	elif t1 <= -4.5e-148:
		tmp = t_1
	elif t1 <= 1.15e-202:
		tmp = v * ((-1.0 / u) * (t1 / u))
	elif t1 <= 1.05e+119:
		tmp = t_1
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -3.4e+123)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= -4.5e-148)
		tmp = t_1;
	elseif (t1 <= 1.15e-202)
		tmp = Float64(v * Float64(Float64(-1.0 / u) * Float64(t1 / u)));
	elseif (t1 <= 1.05e+119)
		tmp = t_1;
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -3.4e+123)
		tmp = -v / (t1 + u);
	elseif (t1 <= -4.5e-148)
		tmp = t_1;
	elseif (t1 <= 1.15e-202)
		tmp = v * ((-1.0 / u) * (t1 / u));
	elseif (t1 <= 1.05e+119)
		tmp = t_1;
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.4e+123], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -4.5e-148], t$95$1, If[LessEqual[t1, 1.15e-202], N[(v * N[(N[(-1.0 / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.05e+119], t$95$1, N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -3.40000000000000001e123

   1. Initial program 39.8%

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

      1. associate-/r* 57.9%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around inf 93.6%

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

      1. neg-mul-1 93.6%

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

   6. Simplified 93.6%

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

   if -3.40000000000000001e123 < t1 < -4.50000000000000015e-148 or 1.1499999999999999e-202 < t1 < 1.04999999999999991e119

   1. Initial program 84.0%

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

      1. associate-*l/ 88.7%

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

      2. *-commutative 88.7%

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

   3. Simplified 88.7%

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

   if -4.50000000000000015e-148 < t1 < 1.1499999999999999e-202

   1. Initial program 75.3%

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

      1. associate-*l/ 82.5%

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

      2. *-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in t1 around 0 82.5%

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

      1. associate-*r/ 82.5%

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

      2. neg-mul-1 82.5%

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

      3. unpow2 82.5%

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

   6. Simplified 82.5%

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

      1. neg-mul-1 82.5%

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

      2. times-frac 90.8%

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

   8. Applied egg-rr 90.8%

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

   if 1.04999999999999991e119 < t1

   1. Initial program 35.8%

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

      1. associate-/r* 60.9%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 92.9%

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

      1. neg-mul-1 92.9%

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

   6. Simplified 92.9%

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

      1. frac-2neg 92.9%

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

      2. div-inv 92.7%

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

      3. remove-double-neg 92.7%

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

      4. +-commutative 92.7%

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

      5. distribute-neg-in 92.7%

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

      6. add-sqr-sqrt 47.7%

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

      7. sqrt-unprod 84.4%

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

      8. sqr-neg 84.4%

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

      9. sqrt-prod 44.9%

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

      10. add-sqr-sqrt 93.0%

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

   8. Applied egg-rr 93.0%

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

      1. sub-neg 93.0%

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

      2. associate-*r/ 93.2%

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

      3. *-rgt-identity 93.2%

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

   10. Simplified 93.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -4.5 \cdot 10^{-148}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.15 \cdot 10^{-202}:\\ \;\;\;\;v \cdot \left(\frac{-1}{u} \cdot \frac{t1}{u}\right)\\ \mathbf{elif}\;t1 \leq 1.05 \cdot 10^{+119}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 3: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.45 \cdot 10^{+16}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.2e-33)
   (/ (- v) (+ t1 u))
   (if (<= t1 3.45e+16) (* v (/ (- t1) (* u u))) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.2e-33) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 3.45e+16) {
		tmp = v * (-t1 / (u * u));
	} else {
		tmp = v / (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 <= (-4.2d-33)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 3.45d+16) then
        tmp = v * (-t1 / (u * u))
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.2e-33) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 3.45e+16) {
		tmp = v * (-t1 / (u * u));
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4.2e-33:
		tmp = -v / (t1 + u)
	elif t1 <= 3.45e+16:
		tmp = v * (-t1 / (u * u))
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.2e-33)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 3.45e+16)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -4.2e-33)
		tmp = -v / (t1 + u);
	elseif (t1 <= 3.45e+16)
		tmp = v * (-t1 / (u * u));
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.2e-33], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.45e+16], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.2e-33

   1. Initial program 56.8%

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

      1. associate-/r* 71.3%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 82.4%

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

      1. neg-mul-1 82.4%

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

   6. Simplified 82.4%

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

   if -4.2e-33 < t1 < 3.45e16

   1. Initial program 79.3%

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

      1. associate-*l/ 84.8%

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

      2. *-commutative 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in t1 around 0 75.5%

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

      1. associate-*r/ 75.5%

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

      2. neg-mul-1 75.5%

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

      3. unpow2 75.5%

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

   6. Simplified 75.5%

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

   if 3.45e16 < t1

   1. Initial program 52.4%

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

      1. associate-/r* 69.8%

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

      2. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t1 around inf 90.3%

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

      1. neg-mul-1 90.3%

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

   6. Simplified 90.3%

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

      1. frac-2neg 90.3%

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

      2. div-inv 90.1%

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

      3. remove-double-neg 90.1%

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

      4. +-commutative 90.1%

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

      5. distribute-neg-in 90.1%

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

      6. add-sqr-sqrt 46.7%

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

      7. sqrt-unprod 87.4%

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

      8. sqr-neg 87.4%

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

      9. sqrt-prod 43.3%

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

      10. add-sqr-sqrt 90.2%

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

   8. Applied egg-rr 90.2%

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

      1. sub-neg 90.2%

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

      2. associate-*r/ 90.4%

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

      3. *-rgt-identity 90.4%

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

   10. Simplified 90.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.45 \cdot 10^{+16}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{-32}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.35e-32)
   (/ (- v) (+ t1 u))
   (if (<= t1 5.6e+20) (* (/ v u) (/ t1 (- u))) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.35e-32) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 5.6e+20) {
		tmp = (v / u) * (t1 / -u);
	} else {
		tmp = v / (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 <= (-1.35d-32)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 5.6d+20) then
        tmp = (v / u) * (t1 / -u)
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.35e-32) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 5.6e+20) {
		tmp = (v / u) * (t1 / -u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.35e-32:
		tmp = -v / (t1 + u)
	elif t1 <= 5.6e+20:
		tmp = (v / u) * (t1 / -u)
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.35e-32)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 5.6e+20)
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.35e-32)
		tmp = -v / (t1 + u);
	elseif (t1 <= 5.6e+20)
		tmp = (v / u) * (t1 / -u);
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.35e-32], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.6e+20], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.3499999999999999e-32

   1. Initial program 56.8%

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

      1. associate-/r* 71.3%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 82.4%

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

      1. neg-mul-1 82.4%

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

   6. Simplified 82.4%

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

   if -1.3499999999999999e-32 < t1 < 5.6e20

   1. Initial program 79.3%

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

      1. associate-*l/ 84.8%

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

      2. *-commutative 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in t1 around 0 75.5%

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

      1. associate-*r/ 75.5%

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

      2. neg-mul-1 75.5%

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

      3. unpow2 75.5%

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

   6. Simplified 75.5%

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

      1. associate-*r/ 73.0%

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

      2. frac-2neg 73.0%

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

      3. add-sqr-sqrt 36.6%

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

      4. sqrt-unprod 49.7%

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

      5. sqr-neg 49.7%

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

      6. sqrt-unprod 21.8%

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

      7. add-sqr-sqrt 41.5%

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

      8. distribute-rgt-neg-out 41.5%

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

      9. add-sqr-sqrt 19.7%

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

      10. sqrt-unprod 50.8%

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

      11. sqr-neg 50.8%

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

      12. sqrt-unprod 36.3%

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

      13. add-sqr-sqrt 73.0%

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

      14. distribute-rgt-neg-in 73.0%

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

   8. Applied egg-rr 73.0%

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

      1. times-frac 83.0%

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

   10. Simplified 83.0%

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

   if 5.6e20 < t1

   1. Initial program 52.4%

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

      1. associate-/r* 69.8%

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

      2. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t1 around inf 90.3%

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

      1. neg-mul-1 90.3%

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

   6. Simplified 90.3%

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

      1. frac-2neg 90.3%

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

      2. div-inv 90.1%

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

      3. remove-double-neg 90.1%

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

      4. +-commutative 90.1%

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

      5. distribute-neg-in 90.1%

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

      6. add-sqr-sqrt 46.7%

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

      7. sqrt-unprod 87.4%

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

      8. sqr-neg 87.4%

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

      9. sqrt-prod 43.3%

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

      10. add-sqr-sqrt 90.2%

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

   8. Applied egg-rr 90.2%

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

      1. sub-neg 90.2%

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

      2. associate-*r/ 90.4%

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

      3. *-rgt-identity 90.4%

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

   10. Simplified 90.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{-32}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 5: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.65 \cdot 10^{+172} \lor \neg \left(u \leq 1.52 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.65e+172) (not (<= u 1.52e+66)))
   (/ v (/ (* u u) t1))
   (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.65e+172) || !(u <= 1.52e+66)) {
		tmp = v / ((u * 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 <= (-2.65d+172)) .or. (.not. (u <= 1.52d+66))) then
        tmp = v / ((u * 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 <= -2.65e+172) || !(u <= 1.52e+66)) {
		tmp = v / ((u * u) / t1);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.65e+172) or not (u <= 1.52e+66):
		tmp = v / ((u * u) / t1)
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.65e+172) || !(u <= 1.52e+66))
		tmp = Float64(v / Float64(Float64(u * 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 <= -2.65e+172) || ~((u <= 1.52e+66)))
		tmp = v / ((u * u) / t1);
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.65e+172], N[Not[LessEqual[u, 1.52e+66]], $MachinePrecision]], N[(v / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -2.65e172 or 1.52000000000000004e66 < u

   1. Initial program 72.9%

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

      1. associate-*l/ 72.4%

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

      2. *-commutative 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in t1 around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. neg-mul-1 69.8%

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

      3. unpow2 69.8%

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

   6. Simplified 69.8%

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

      1. clear-num 69.8%

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

      2. un-div-inv 69.8%

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

      3. add-sqr-sqrt 29.5%

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

      4. sqrt-unprod 57.9%

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

      5. sqr-neg 57.9%

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

      6. sqrt-unprod 37.6%

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

      7. add-sqr-sqrt 65.8%

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

   8. Applied egg-rr 65.8%

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

   if -2.65e172 < u < 1.52000000000000004e66

   1. Initial program 64.0%

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

      1. associate-*l/ 70.5%

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

      2. *-commutative 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in t1 around inf 71.0%

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

      1. associate-*r/ 71.0%

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

      2. neg-mul-1 71.0%

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

   6. Simplified 71.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.65 \cdot 10^{+172} \lor \neg \left(u \leq 1.52 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 6: 23.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{+83}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.05 \cdot 10^{+39}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.12e+83) (/ v t1) (if (<= t1 1.05e+39) (/ v u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.12e+83) {
		tmp = v / t1;
	} else if (t1 <= 1.05e+39) {
		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 <= (-1.12d+83)) then
        tmp = v / t1
    else if (t1 <= 1.05d+39) 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 <= -1.12e+83) {
		tmp = v / t1;
	} else if (t1 <= 1.05e+39) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.12e+83:
		tmp = v / t1
	elif t1 <= 1.05e+39:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.12e+83)
		tmp = Float64(v / t1);
	elseif (t1 <= 1.05e+39)
		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 <= -1.12e+83)
		tmp = v / t1;
	elseif (t1 <= 1.05e+39)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.12e+83], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 1.05e+39], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.12e83 or 1.0499999999999999e39 < t1

   1. Initial program 50.3%

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

      1. associate-/r* 67.1%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 92.2%

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

      1. neg-mul-1 92.2%

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

   6. Simplified 92.2%

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

      1. clear-num 91.7%

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

      2. associate-/r/ 92.0%

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

      3. add-sqr-sqrt 49.5%

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

      4. sqrt-unprod 55.7%

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

      5. sqr-neg 55.7%

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

      6. sqrt-unprod 13.2%

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

      7. add-sqr-sqrt 29.7%

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

   8. Applied egg-rr 29.7%

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

   9. Taylor expanded in t1 around inf 28.6%

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

   if -1.12e83 < t1 < 1.0499999999999999e39

   1. Initial program 78.4%

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

      1. associate-/r* 87.7%

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

      2. associate-/l* 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in t1 around inf 38.9%

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

      1. neg-mul-1 38.9%

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

   6. Simplified 38.9%

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

      1. frac-2neg 38.9%

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

      2. div-inv 38.8%

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

      3. remove-double-neg 38.8%

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

      4. +-commutative 38.8%

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

      5. distribute-neg-in 38.8%

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

      6. add-sqr-sqrt 17.5%

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

      7. sqrt-unprod 50.4%

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

      8. sqr-neg 50.4%

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

      9. sqrt-prod 22.1%

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

      10. add-sqr-sqrt 39.4%

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

   8. Applied egg-rr 39.4%

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

      1. sub-neg 39.4%

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

      2. associate-*r/ 39.4%

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

      3. *-rgt-identity 39.4%

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

   10. Simplified 39.4%

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

   11. Taylor expanded in u around inf 15.9%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{+83}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.05 \cdot 10^{+39}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 7: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 8.2 \cdot 10^{+177}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= u 8.2e+177) (- (/ v t1)) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 8.2e+177) {
		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 <= 8.2d+177) 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 <= 8.2e+177) {
		tmp = -(v / t1);
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 8.2e+177:
		tmp = -(v / t1)
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 8.2e+177)
		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 <= 8.2e+177)
		tmp = -(v / t1);
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 8.2e+177], (-N[(v / t1), $MachinePrecision]), N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 8.20000000000000029e177

   1. Initial program 66.5%

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

      1. associate-*l/ 71.3%

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

      2. *-commutative 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in t1 around inf 60.3%

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

      1. associate-*r/ 60.3%

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

      2. neg-mul-1 60.3%

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

   6. Simplified 60.3%

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

   if 8.20000000000000029e177 < u

   1. Initial program 68.3%

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

      1. associate-/r* 91.6%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 44.3%

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

      1. neg-mul-1 44.3%

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

   6. Simplified 44.3%

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

      1. frac-2neg 44.3%

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

      2. div-inv 44.3%

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

      3. remove-double-neg 44.3%

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

      4. +-commutative 44.3%

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

      5. distribute-neg-in 44.3%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 68.8%

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

      8. sqr-neg 68.8%

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

      9. sqrt-prod 44.4%

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

      10. add-sqr-sqrt 44.4%

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

   8. Applied egg-rr 44.4%

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

      1. sub-neg 44.4%

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

      2. associate-*r/ 44.4%

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

      3. *-rgt-identity 44.4%

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

   10. Simplified 44.4%

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

   11. Taylor expanded in u around inf 40.4%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 8.2 \cdot 10^{+177}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 8: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 1.16 \cdot 10^{+178}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 1.16e+178) (- (/ v t1)) (- (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.16e+178) {
		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 <= 1.16d+178) 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 <= 1.16e+178) {
		tmp = -(v / t1);
	} else {
		tmp = -(v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 1.16e+178:
		tmp = -(v / t1)
	else:
		tmp = -(v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 1.16e+178)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(-Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 1.16e+178)
		tmp = -(v / t1);
	else
		tmp = -(v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 1.16e+178], (-N[(v / t1), $MachinePrecision]), (-N[(v / u), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < 1.16000000000000005e178

   1. Initial program 66.5%

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

      1. associate-*l/ 71.3%

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

      2. *-commutative 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in t1 around inf 60.3%

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

      1. associate-*r/ 60.3%

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

      2. neg-mul-1 60.3%

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

   6. Simplified 60.3%

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

   if 1.16000000000000005e178 < u

   1. Initial program 68.3%

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

      1. associate-/r* 91.6%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 44.0%

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

   5. Taylor expanded in t1 around 0 40.4%

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

      1. associate-*r/ 40.4%

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

      2. neg-mul-1 40.4%

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

   7. Simplified 40.4%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 1.16 \cdot 10^{+178}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{u}\\ \end{array} \]

Alternative 9: 62.4% accurate, 2.4× 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(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 66.6%

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

   1. associate-/r* 79.1%

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

   2. associate-/l* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in t1 around inf 61.2%

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

   1. neg-mul-1 61.2%

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

6. Simplified 61.2%

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

   1. frac-2neg 61.2%

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

   2. div-inv 61.0%

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

   3. remove-double-neg 61.0%

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

   4. +-commutative 61.0%

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

   5. distribute-neg-in 61.0%

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

   6. add-sqr-sqrt 28.6%

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

   7. sqrt-unprod 66.8%

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

   8. sqr-neg 66.8%

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

   9. sqrt-prod 32.8%

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

   10. add-sqr-sqrt 61.3%

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

8. Applied egg-rr 61.3%

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

   1. sub-neg 61.3%

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

   2. associate-*r/ 61.5%

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

   3. *-rgt-identity 61.5%

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

10. Simplified 61.5%

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

11. Final simplification 61.5%

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

Alternative 10: 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 66.6%

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

   1. associate-/r* 79.1%

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

   2. associate-/l* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in t1 around inf 61.2%

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

   1. neg-mul-1 61.2%

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

6. Simplified 61.2%

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

   1. clear-num 61.2%

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

   2. associate-/r/ 61.0%

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

   3. add-sqr-sqrt 30.3%

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

   4. sqrt-unprod 39.7%

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

   5. sqr-neg 39.7%

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

   6. sqrt-unprod 9.7%

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

   7. add-sqr-sqrt 20.2%

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

8. Applied egg-rr 20.2%

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

9. Taylor expanded in t1 around inf 13.1%

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

10. Final simplification 13.1%

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

Reproduce

herbie shell --seed 2023274 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))