Rosa's DopplerBench

Percentage Accurate: 72.6% → 98.0%

Time: 11.2s

Alternatives: 14

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: 72.6% 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 74.2%

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

   1. times-frac 97.0%

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

3. Simplified 97.0%

   \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing

5. Final simplification 97.0%

   \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u} \]
6. Add Preprocessing

Alternative 2: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ t_2 := \frac{\frac{t1}{\frac{t1 + u}{v}}}{t1 - u}\\ \mathbf{if}\;u \leq -0.0033:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 2.9 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 0.0072:\\ \;\;\;\;v \cdot \frac{\frac{t1}{t1 - u}}{t1 + u}\\ \mathbf{elif}\;u \leq 3.9 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))) (t_2 (/ (/ t1 (/ (+ t1 u) v)) (- t1 u))))
   (if (<= u -0.0033)
     t_2
     (if (<= u 2.9e-38)
       t_1
       (if (<= u 0.0072)
         (* v (/ (/ t1 (- t1 u)) (+ t1 u)))
         (if (<= u 3.9e+64) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double t_2 = (t1 / ((t1 + u) / v)) / (t1 - u);
	double tmp;
	if (u <= -0.0033) {
		tmp = t_2;
	} else if (u <= 2.9e-38) {
		tmp = t_1;
	} else if (u <= 0.0072) {
		tmp = v * ((t1 / (t1 - u)) / (t1 + u));
	} else if (u <= 3.9e+64) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    t_2 = (t1 / ((t1 + u) / v)) / (t1 - u)
    if (u <= (-0.0033d0)) then
        tmp = t_2
    else if (u <= 2.9d-38) then
        tmp = t_1
    else if (u <= 0.0072d0) then
        tmp = v * ((t1 / (t1 - u)) / (t1 + u))
    else if (u <= 3.9d+64) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double t_2 = (t1 / ((t1 + u) / v)) / (t1 - u);
	double tmp;
	if (u <= -0.0033) {
		tmp = t_2;
	} else if (u <= 2.9e-38) {
		tmp = t_1;
	} else if (u <= 0.0072) {
		tmp = v * ((t1 / (t1 - u)) / (t1 + u));
	} else if (u <= 3.9e+64) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	t_2 = (t1 / ((t1 + u) / v)) / (t1 - u)
	tmp = 0
	if u <= -0.0033:
		tmp = t_2
	elif u <= 2.9e-38:
		tmp = t_1
	elif u <= 0.0072:
		tmp = v * ((t1 / (t1 - u)) / (t1 + u))
	elif u <= 3.9e+64:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	t_2 = Float64(Float64(t1 / Float64(Float64(t1 + u) / v)) / Float64(t1 - u))
	tmp = 0.0
	if (u <= -0.0033)
		tmp = t_2;
	elseif (u <= 2.9e-38)
		tmp = t_1;
	elseif (u <= 0.0072)
		tmp = Float64(v * Float64(Float64(t1 / Float64(t1 - u)) / Float64(t1 + u)));
	elseif (u <= 3.9e+64)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	t_2 = (t1 / ((t1 + u) / v)) / (t1 - u);
	tmp = 0.0;
	if (u <= -0.0033)
		tmp = t_2;
	elseif (u <= 2.9e-38)
		tmp = t_1;
	elseif (u <= 0.0072)
		tmp = v * ((t1 / (t1 - u)) / (t1 + u));
	elseif (u <= 3.9e+64)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t1 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -0.0033], t$95$2, If[LessEqual[u, 2.9e-38], t$95$1, If[LessEqual[u, 0.0072], N[(v * N[(N[(t1 / N[(t1 - u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.9e+64], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -0.0033 or 3.8999999999999998e64 < u

   1. Initial program 78.9%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

      1. *-commutative 98.0%

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

      2. clear-num 97.9%

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

      3. frac-2neg 97.9%

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

      4. frac-times 88.0%

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

      5. *-un-lft-identity 88.0%

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

      6. remove-double-neg 88.0%

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

      7. distribute-neg-in 88.0%

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

      8. add-sqr-sqrt 44.7%

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

      9. sqrt-unprod 83.5%

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

      10. sqr-neg 83.5%

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

      11. sqrt-unprod 41.0%

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

      12. add-sqr-sqrt 83.4%

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

      13. sub-neg 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate-/r* 86.2%

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

   8. Simplified 86.2%

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

   if -0.0033 < u < 2.89999999999999994e-38 or 0.0071999999999999998 < u < 3.8999999999999998e64

   1. Initial program 67.7%

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

      1. associate-/r* 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-/l* 99.0%

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

      4. associate-/l/ 99.1%

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

      5. +-commutative 99.1%

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

      6. remove-double-neg 99.1%

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

      7. unsub-neg 99.1%

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

      8. div-sub 99.1%

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

      9. sub-neg 99.1%

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

      10. *-inverses 99.1%

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

      11. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.0%

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

      1. mul-1-neg 86.0%

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

      2. unsub-neg 86.0%

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

      3. *-commutative 86.0%

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

   7. Simplified 86.0%

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

   if 2.89999999999999994e-38 < u < 0.0071999999999999998

   1. Initial program 99.0%

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

      1. times-frac 64.5%

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

   3. Simplified 64.5%

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

      1. *-commutative 64.5%

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

      2. clear-num 64.5%

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

      3. frac-2neg 64.5%

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

      4. frac-times 59.4%

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

      5. *-un-lft-identity 59.4%

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

      6. remove-double-neg 59.4%

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

      7. distribute-neg-in 59.4%

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

      8. add-sqr-sqrt 38.7%

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

      9. sqrt-unprod 59.4%

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

      10. sqr-neg 59.4%

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

      11. sqrt-unprod 20.7%

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

      12. add-sqr-sqrt 59.4%

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

      13. sub-neg 59.4%

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

   6. Applied egg-rr 59.4%

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

      1. associate-/r* 70.2%

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

   8. Simplified 70.2%

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

      1. div-inv 64.7%

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

      2. clear-num 64.7%

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

      3. sub-neg 64.7%

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

      4. add-sqr-sqrt 26.0%

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

      5. sqrt-unprod 64.7%

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

      6. sqr-neg 64.7%

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

      7. sqrt-unprod 38.7%

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

      8. add-sqr-sqrt 64.7%

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

      9. distribute-neg-in 64.7%

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

      10. associate-*l/ 64.5%

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

      11. remove-double-neg 64.5%

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

      12. distribute-frac-neg 64.5%

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

      13. frac-2neg 64.5%

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

      14. distribute-frac-neg 64.5%

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

      15. clear-num 64.5%

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

      16. div-inv 70.0%

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

      17. associate-/r/ 99.6%

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

   10. Applied egg-rr 99.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.0033:\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 + u}{v}}}{t1 - u}\\ \mathbf{elif}\;u \leq 2.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq 0.0072:\\ \;\;\;\;v \cdot \frac{\frac{t1}{t1 - u}}{t1 + u}\\ \mathbf{elif}\;u \leq 3.9 \cdot 10^{+64}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 + u}{v}}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ t_2 := t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \mathbf{if}\;u \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{v}{\frac{t1 - u}{\frac{t1}{u}}}\\ \mathbf{elif}\;u \leq 1.95 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))) (t_2 (* t1 (/ (/ v u) (- t1 u)))))
   (if (<= u -1.35e-6)
     t_2
     (if (<= u 3.8e-37)
       t_1
       (if (<= u 4.5e-7)
         (/ v (/ (- t1 u) (/ t1 u)))
         (if (<= u 1.95e+63) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double t_2 = t1 * ((v / u) / (t1 - u));
	double tmp;
	if (u <= -1.35e-6) {
		tmp = t_2;
	} else if (u <= 3.8e-37) {
		tmp = t_1;
	} else if (u <= 4.5e-7) {
		tmp = v / ((t1 - u) / (t1 / u));
	} else if (u <= 1.95e+63) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    t_2 = t1 * ((v / u) / (t1 - u))
    if (u <= (-1.35d-6)) then
        tmp = t_2
    else if (u <= 3.8d-37) then
        tmp = t_1
    else if (u <= 4.5d-7) then
        tmp = v / ((t1 - u) / (t1 / u))
    else if (u <= 1.95d+63) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double t_2 = t1 * ((v / u) / (t1 - u));
	double tmp;
	if (u <= -1.35e-6) {
		tmp = t_2;
	} else if (u <= 3.8e-37) {
		tmp = t_1;
	} else if (u <= 4.5e-7) {
		tmp = v / ((t1 - u) / (t1 / u));
	} else if (u <= 1.95e+63) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	t_2 = t1 * ((v / u) / (t1 - u))
	tmp = 0
	if u <= -1.35e-6:
		tmp = t_2
	elif u <= 3.8e-37:
		tmp = t_1
	elif u <= 4.5e-7:
		tmp = v / ((t1 - u) / (t1 / u))
	elif u <= 1.95e+63:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	t_2 = Float64(t1 * Float64(Float64(v / u) / Float64(t1 - u)))
	tmp = 0.0
	if (u <= -1.35e-6)
		tmp = t_2;
	elseif (u <= 3.8e-37)
		tmp = t_1;
	elseif (u <= 4.5e-7)
		tmp = Float64(v / Float64(Float64(t1 - u) / Float64(t1 / u)));
	elseif (u <= 1.95e+63)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	t_2 = t1 * ((v / u) / (t1 - u));
	tmp = 0.0;
	if (u <= -1.35e-6)
		tmp = t_2;
	elseif (u <= 3.8e-37)
		tmp = t_1;
	elseif (u <= 4.5e-7)
		tmp = v / ((t1 - u) / (t1 / u));
	elseif (u <= 1.95e+63)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t1 * N[(N[(v / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.35e-6], t$95$2, If[LessEqual[u, 3.8e-37], t$95$1, If[LessEqual[u, 4.5e-7], N[(v / N[(N[(t1 - u), $MachinePrecision] / N[(t1 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.95e+63], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.34999999999999999e-6 or 1.95e63 < u

   1. Initial program 78.9%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

      1. *-commutative 98.0%

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

      2. clear-num 97.9%

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

      3. frac-2neg 97.9%

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

      4. frac-times 88.0%

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

      5. *-un-lft-identity 88.0%

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

      6. remove-double-neg 88.0%

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

      7. distribute-neg-in 88.0%

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

      8. add-sqr-sqrt 44.7%

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

      9. sqrt-unprod 83.5%

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

      10. sqr-neg 83.5%

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

      11. sqrt-unprod 41.0%

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

      12. add-sqr-sqrt 83.4%

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

      13. sub-neg 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate-/r* 86.2%

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

   8. Simplified 86.2%

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

   9. Taylor expanded in t1 around 0 82.6%

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

   10. Taylor expanded in v around 0 72.9%

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

       1. associate-/r* 77.8%

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

       2. associate-*r/ 82.5%

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

       3. associate-*r/ 83.6%

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

   12. Simplified 83.6%

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

   if -1.34999999999999999e-6 < u < 3.8000000000000004e-37 or 4.4999999999999998e-7 < u < 1.95e63

   1. Initial program 67.7%

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

      1. associate-/r* 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-/l* 99.0%

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

      4. associate-/l/ 99.1%

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

      5. +-commutative 99.1%

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

      6. remove-double-neg 99.1%

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

      7. unsub-neg 99.1%

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

      8. div-sub 99.1%

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

      9. sub-neg 99.1%

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

      10. *-inverses 99.1%

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

      11. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.0%

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

      1. mul-1-neg 86.0%

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

      2. unsub-neg 86.0%

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

      3. *-commutative 86.0%

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

   7. Simplified 86.0%

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

   if 3.8000000000000004e-37 < u < 4.4999999999999998e-7

   1. Initial program 99.0%

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

      1. times-frac 64.5%

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

   3. Simplified 64.5%

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

      1. *-commutative 64.5%

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

      2. clear-num 64.5%

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

      3. frac-2neg 64.5%

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

      4. frac-times 59.4%

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

      5. *-un-lft-identity 59.4%

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

      6. remove-double-neg 59.4%

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

      7. distribute-neg-in 59.4%

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

      8. add-sqr-sqrt 38.7%

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

      9. sqrt-unprod 59.4%

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

      10. sqr-neg 59.4%

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

      11. sqrt-unprod 20.7%

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

      12. add-sqr-sqrt 59.4%

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

      13. sub-neg 59.4%

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

   6. Applied egg-rr 59.4%

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

      1. associate-/r* 70.2%

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

   8. Simplified 70.2%

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

   9. Taylor expanded in t1 around 0 58.4%

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

   10. Taylor expanded in v around 0 87.2%

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

       1. associate-/r* 87.6%

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

       2. associate-*r/ 52.8%

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

       3. associate-*r/ 52.5%

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

   12. Simplified 52.5%

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

       1. associate-/l/ 52.7%

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

       2. associate-*r/ 87.2%

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

       3. associate-/l/ 87.6%

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

       4. associate-*l/ 88.0%

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

       5. *-commutative 88.0%

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

       6. associate-/l* 87.8%

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

   14. Applied egg-rr 87.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{v}{\frac{t1 - u}{\frac{t1}{u}}}\\ \mathbf{elif}\;u \leq 1.95 \cdot 10^{+63}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ t_2 := t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \mathbf{if}\;u \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 0.01:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))) (t_2 (* t1 (/ (/ v u) (- t1 u)))))
   (if (<= u -2.6e-5)
     t_2
     (if (<= u 2.8e-37)
       t_1
       (if (<= u 0.01)
         (/ (* v (/ t1 u)) (- t1 u))
         (if (<= u 2.8e+63) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double t_2 = t1 * ((v / u) / (t1 - u));
	double tmp;
	if (u <= -2.6e-5) {
		tmp = t_2;
	} else if (u <= 2.8e-37) {
		tmp = t_1;
	} else if (u <= 0.01) {
		tmp = (v * (t1 / u)) / (t1 - u);
	} else if (u <= 2.8e+63) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    t_2 = t1 * ((v / u) / (t1 - u))
    if (u <= (-2.6d-5)) then
        tmp = t_2
    else if (u <= 2.8d-37) then
        tmp = t_1
    else if (u <= 0.01d0) then
        tmp = (v * (t1 / u)) / (t1 - u)
    else if (u <= 2.8d+63) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double t_2 = t1 * ((v / u) / (t1 - u));
	double tmp;
	if (u <= -2.6e-5) {
		tmp = t_2;
	} else if (u <= 2.8e-37) {
		tmp = t_1;
	} else if (u <= 0.01) {
		tmp = (v * (t1 / u)) / (t1 - u);
	} else if (u <= 2.8e+63) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	t_2 = t1 * ((v / u) / (t1 - u))
	tmp = 0
	if u <= -2.6e-5:
		tmp = t_2
	elif u <= 2.8e-37:
		tmp = t_1
	elif u <= 0.01:
		tmp = (v * (t1 / u)) / (t1 - u)
	elif u <= 2.8e+63:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	t_2 = Float64(t1 * Float64(Float64(v / u) / Float64(t1 - u)))
	tmp = 0.0
	if (u <= -2.6e-5)
		tmp = t_2;
	elseif (u <= 2.8e-37)
		tmp = t_1;
	elseif (u <= 0.01)
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(t1 - u));
	elseif (u <= 2.8e+63)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	t_2 = t1 * ((v / u) / (t1 - u));
	tmp = 0.0;
	if (u <= -2.6e-5)
		tmp = t_2;
	elseif (u <= 2.8e-37)
		tmp = t_1;
	elseif (u <= 0.01)
		tmp = (v * (t1 / u)) / (t1 - u);
	elseif (u <= 2.8e+63)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t1 * N[(N[(v / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.6e-5], t$95$2, If[LessEqual[u, 2.8e-37], t$95$1, If[LessEqual[u, 0.01], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.8e+63], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.59999999999999984e-5 or 2.79999999999999987e63 < u

   1. Initial program 78.9%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

      1. *-commutative 98.0%

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

      2. clear-num 97.9%

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

      3. frac-2neg 97.9%

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

      4. frac-times 88.0%

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

      5. *-un-lft-identity 88.0%

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

      6. remove-double-neg 88.0%

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

      7. distribute-neg-in 88.0%

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

      8. add-sqr-sqrt 44.7%

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

      9. sqrt-unprod 83.5%

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

      10. sqr-neg 83.5%

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

      11. sqrt-unprod 41.0%

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

      12. add-sqr-sqrt 83.4%

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

      13. sub-neg 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate-/r* 86.2%

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

   8. Simplified 86.2%

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

   9. Taylor expanded in t1 around 0 82.6%

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

   10. Taylor expanded in v around 0 72.9%

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

       1. associate-/r* 77.8%

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

       2. associate-*r/ 82.5%

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

       3. associate-*r/ 83.6%

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

   12. Simplified 83.6%

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

   if -2.59999999999999984e-5 < u < 2.8000000000000001e-37 or 0.0100000000000000002 < u < 2.79999999999999987e63

   1. Initial program 67.7%

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

      1. associate-/r* 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-/l* 99.0%

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

      4. associate-/l/ 99.1%

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

      5. +-commutative 99.1%

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

      6. remove-double-neg 99.1%

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

      7. unsub-neg 99.1%

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

      8. div-sub 99.1%

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

      9. sub-neg 99.1%

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

      10. *-inverses 99.1%

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

      11. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.0%

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

      1. mul-1-neg 86.0%

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

      2. unsub-neg 86.0%

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

      3. *-commutative 86.0%

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

   7. Simplified 86.0%

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

   if 2.8000000000000001e-37 < u < 0.0100000000000000002

   1. Initial program 99.0%

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

      1. times-frac 64.5%

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

   3. Simplified 64.5%

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

      1. *-commutative 64.5%

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

      2. clear-num 64.5%

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

      3. frac-2neg 64.5%

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

      4. frac-times 59.4%

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

      5. *-un-lft-identity 59.4%

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

      6. remove-double-neg 59.4%

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

      7. distribute-neg-in 59.4%

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

      8. add-sqr-sqrt 38.7%

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

      9. sqrt-unprod 59.4%

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

      10. sqr-neg 59.4%

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

      11. sqrt-unprod 20.7%

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

      12. add-sqr-sqrt 59.4%

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

      13. sub-neg 59.4%

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

   6. Applied egg-rr 59.4%

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

      1. associate-/r* 70.2%

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

   8. Simplified 70.2%

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

   9. Taylor expanded in t1 around 0 58.4%

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

       1. associate-/r/ 88.0%

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

   11. Applied egg-rr 88.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq 0.01:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{+63}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;u \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 2.6 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 0.00106:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= u -1.15e-7)
     (* (- t1) (/ (/ v u) (+ t1 u)))
     (if (<= u 2.6e-37)
       t_1
       (if (<= u 0.00106)
         (/ (* v (/ t1 u)) (- t1 u))
         (if (<= u 2.8e+63) t_1 (* t1 (/ (/ v u) (- t1 u)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (u <= -1.15e-7) {
		tmp = -t1 * ((v / u) / (t1 + u));
	} else if (u <= 2.6e-37) {
		tmp = t_1;
	} else if (u <= 0.00106) {
		tmp = (v * (t1 / u)) / (t1 - u);
	} else if (u <= 2.8e+63) {
		tmp = t_1;
	} else {
		tmp = t1 * ((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) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (u <= (-1.15d-7)) then
        tmp = -t1 * ((v / u) / (t1 + u))
    else if (u <= 2.6d-37) then
        tmp = t_1
    else if (u <= 0.00106d0) then
        tmp = (v * (t1 / u)) / (t1 - u)
    else if (u <= 2.8d+63) then
        tmp = t_1
    else
        tmp = t1 * ((v / u) / (t1 - u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (u <= -1.15e-7) {
		tmp = -t1 * ((v / u) / (t1 + u));
	} else if (u <= 2.6e-37) {
		tmp = t_1;
	} else if (u <= 0.00106) {
		tmp = (v * (t1 / u)) / (t1 - u);
	} else if (u <= 2.8e+63) {
		tmp = t_1;
	} else {
		tmp = t1 * ((v / u) / (t1 - u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if u <= -1.15e-7:
		tmp = -t1 * ((v / u) / (t1 + u))
	elif u <= 2.6e-37:
		tmp = t_1
	elif u <= 0.00106:
		tmp = (v * (t1 / u)) / (t1 - u)
	elif u <= 2.8e+63:
		tmp = t_1
	else:
		tmp = t1 * ((v / u) / (t1 - u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (u <= -1.15e-7)
		tmp = Float64(Float64(-t1) * Float64(Float64(v / u) / Float64(t1 + u)));
	elseif (u <= 2.6e-37)
		tmp = t_1;
	elseif (u <= 0.00106)
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(t1 - u));
	elseif (u <= 2.8e+63)
		tmp = t_1;
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(t1 - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (u <= -1.15e-7)
		tmp = -t1 * ((v / u) / (t1 + u));
	elseif (u <= 2.6e-37)
		tmp = t_1;
	elseif (u <= 0.00106)
		tmp = (v * (t1 / u)) / (t1 - u);
	elseif (u <= 2.8e+63)
		tmp = t_1;
	else
		tmp = t1 * ((v / u) / (t1 - u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.15e-7], N[((-t1) * N[(N[(v / u), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.6e-37], t$95$1, If[LessEqual[u, 0.00106], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.8e+63], t$95$1, N[(t1 * N[(N[(v / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.14999999999999997e-7

   1. Initial program 80.0%

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

      1. times-frac 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 78.9%

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

   6. Taylor expanded in v around 0 69.0%

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

      1. associate-*r/ 69.0%

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

      2. neg-mul-1 69.0%

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

      3. +-commutative 69.0%

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

      4. *-commutative 69.0%

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

      5. distribute-frac-neg 69.0%

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

      6. times-frac 78.9%

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

      7. associate-*l/ 80.2%

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

      8. associate-*r/ 81.9%

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

      9. distribute-rgt-neg-in 81.9%

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

      10. +-commutative 81.9%

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

   8. Simplified 81.9%

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

   if -1.14999999999999997e-7 < u < 2.5999999999999998e-37 or 0.00105999999999999996 < u < 2.79999999999999987e63

   1. Initial program 67.7%

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

      1. associate-/r* 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-/l* 99.0%

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

      4. associate-/l/ 99.1%

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

      5. +-commutative 99.1%

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

      6. remove-double-neg 99.1%

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

      7. unsub-neg 99.1%

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

      8. div-sub 99.1%

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

      9. sub-neg 99.1%

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

      10. *-inverses 99.1%

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

      11. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.0%

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

      1. mul-1-neg 86.0%

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

      2. unsub-neg 86.0%

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

      3. *-commutative 86.0%

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

   7. Simplified 86.0%

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

   if 2.5999999999999998e-37 < u < 0.00105999999999999996

   1. Initial program 99.0%

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

      1. times-frac 64.5%

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

   3. Simplified 64.5%

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

      1. *-commutative 64.5%

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

      2. clear-num 64.5%

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

      3. frac-2neg 64.5%

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

      4. frac-times 59.4%

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

      5. *-un-lft-identity 59.4%

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

      6. remove-double-neg 59.4%

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

      7. distribute-neg-in 59.4%

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

      8. add-sqr-sqrt 38.7%

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

      9. sqrt-unprod 59.4%

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

      10. sqr-neg 59.4%

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

      11. sqrt-unprod 20.7%

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

      12. add-sqr-sqrt 59.4%

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

      13. sub-neg 59.4%

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

   6. Applied egg-rr 59.4%

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

      1. associate-/r* 70.2%

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

   8. Simplified 70.2%

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

   9. Taylor expanded in t1 around 0 58.4%

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

       1. associate-/r/ 88.0%

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

   11. Applied egg-rr 88.0%

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

   if 2.79999999999999987e63 < u

   1. Initial program 77.7%

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

      1. times-frac 97.4%

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

   3. Simplified 97.4%

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

      1. *-commutative 97.4%

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

      2. clear-num 97.4%

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

      3. frac-2neg 97.4%

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

      4. frac-times 87.0%

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

      5. *-un-lft-identity 87.0%

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

      6. remove-double-neg 87.0%

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

      7. distribute-neg-in 87.0%

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

      8. add-sqr-sqrt 36.5%

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

      9. sqrt-unprod 83.8%

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

      10. sqr-neg 83.8%

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

      11. sqrt-unprod 48.8%

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

      12. add-sqr-sqrt 85.4%

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

      13. sub-neg 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-/r* 88.3%

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

   8. Simplified 88.3%

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

   9. Taylor expanded in t1 around 0 85.3%

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

   10. Taylor expanded in v around 0 77.6%

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

       1. associate-/r* 80.2%

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

       2. associate-*r/ 85.3%

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

       3. associate-*r/ 85.8%

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

   12. Simplified 85.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 2.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq 0.00106:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{+63}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;u \leq -0.00052:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 1.76 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 0.66:\\ \;\;\;\;v \cdot \frac{\frac{t1}{t1 - u}}{t1 + u}\\ \mathbf{elif}\;u \leq 1.95 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= u -0.00052)
     (* (- t1) (/ (/ v u) (+ t1 u)))
     (if (<= u 1.76e-37)
       t_1
       (if (<= u 0.66)
         (* v (/ (/ t1 (- t1 u)) (+ t1 u)))
         (if (<= u 1.95e+63) t_1 (* t1 (/ (/ v u) (- t1 u)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (u <= -0.00052) {
		tmp = -t1 * ((v / u) / (t1 + u));
	} else if (u <= 1.76e-37) {
		tmp = t_1;
	} else if (u <= 0.66) {
		tmp = v * ((t1 / (t1 - u)) / (t1 + u));
	} else if (u <= 1.95e+63) {
		tmp = t_1;
	} else {
		tmp = t1 * ((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) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (u <= (-0.00052d0)) then
        tmp = -t1 * ((v / u) / (t1 + u))
    else if (u <= 1.76d-37) then
        tmp = t_1
    else if (u <= 0.66d0) then
        tmp = v * ((t1 / (t1 - u)) / (t1 + u))
    else if (u <= 1.95d+63) then
        tmp = t_1
    else
        tmp = t1 * ((v / u) / (t1 - u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (u <= -0.00052) {
		tmp = -t1 * ((v / u) / (t1 + u));
	} else if (u <= 1.76e-37) {
		tmp = t_1;
	} else if (u <= 0.66) {
		tmp = v * ((t1 / (t1 - u)) / (t1 + u));
	} else if (u <= 1.95e+63) {
		tmp = t_1;
	} else {
		tmp = t1 * ((v / u) / (t1 - u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if u <= -0.00052:
		tmp = -t1 * ((v / u) / (t1 + u))
	elif u <= 1.76e-37:
		tmp = t_1
	elif u <= 0.66:
		tmp = v * ((t1 / (t1 - u)) / (t1 + u))
	elif u <= 1.95e+63:
		tmp = t_1
	else:
		tmp = t1 * ((v / u) / (t1 - u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (u <= -0.00052)
		tmp = Float64(Float64(-t1) * Float64(Float64(v / u) / Float64(t1 + u)));
	elseif (u <= 1.76e-37)
		tmp = t_1;
	elseif (u <= 0.66)
		tmp = Float64(v * Float64(Float64(t1 / Float64(t1 - u)) / Float64(t1 + u)));
	elseif (u <= 1.95e+63)
		tmp = t_1;
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(t1 - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (u <= -0.00052)
		tmp = -t1 * ((v / u) / (t1 + u));
	elseif (u <= 1.76e-37)
		tmp = t_1;
	elseif (u <= 0.66)
		tmp = v * ((t1 / (t1 - u)) / (t1 + u));
	elseif (u <= 1.95e+63)
		tmp = t_1;
	else
		tmp = t1 * ((v / u) / (t1 - u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -0.00052], N[((-t1) * N[(N[(v / u), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.76e-37], t$95$1, If[LessEqual[u, 0.66], N[(v * N[(N[(t1 / N[(t1 - u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.95e+63], t$95$1, N[(t1 * N[(N[(v / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -5.19999999999999954e-4

   1. Initial program 80.0%

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

      1. times-frac 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 78.9%

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

   6. Taylor expanded in v around 0 69.0%

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

      1. associate-*r/ 69.0%

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

      2. neg-mul-1 69.0%

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

      3. +-commutative 69.0%

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

      4. *-commutative 69.0%

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

      5. distribute-frac-neg 69.0%

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

      6. times-frac 78.9%

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

      7. associate-*l/ 80.2%

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

      8. associate-*r/ 81.9%

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

      9. distribute-rgt-neg-in 81.9%

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

      10. +-commutative 81.9%

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

   8. Simplified 81.9%

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

   if -5.19999999999999954e-4 < u < 1.76000000000000006e-37 or 0.660000000000000031 < u < 1.95e63

   1. Initial program 67.7%

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

      1. associate-/r* 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-/l* 99.0%

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

      4. associate-/l/ 99.1%

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

      5. +-commutative 99.1%

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

      6. remove-double-neg 99.1%

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

      7. unsub-neg 99.1%

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

      8. div-sub 99.1%

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

      9. sub-neg 99.1%

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

      10. *-inverses 99.1%

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

      11. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.0%

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

      1. mul-1-neg 86.0%

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

      2. unsub-neg 86.0%

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

      3. *-commutative 86.0%

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

   7. Simplified 86.0%

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

   if 1.76000000000000006e-37 < u < 0.660000000000000031

   1. Initial program 99.0%

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

      1. times-frac 64.5%

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

   3. Simplified 64.5%

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

      1. *-commutative 64.5%

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

      2. clear-num 64.5%

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

      3. frac-2neg 64.5%

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

      4. frac-times 59.4%

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

      5. *-un-lft-identity 59.4%

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

      6. remove-double-neg 59.4%

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

      7. distribute-neg-in 59.4%

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

      8. add-sqr-sqrt 38.7%

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

      9. sqrt-unprod 59.4%

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

      10. sqr-neg 59.4%

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

      11. sqrt-unprod 20.7%

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

      12. add-sqr-sqrt 59.4%

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

      13. sub-neg 59.4%

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

   6. Applied egg-rr 59.4%

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

      1. associate-/r* 70.2%

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

   8. Simplified 70.2%

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

      1. div-inv 64.7%

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

      2. clear-num 64.7%

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

      3. sub-neg 64.7%

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

      4. add-sqr-sqrt 26.0%

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

      5. sqrt-unprod 64.7%

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

      6. sqr-neg 64.7%

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

      7. sqrt-unprod 38.7%

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

      8. add-sqr-sqrt 64.7%

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

      9. distribute-neg-in 64.7%

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

      10. associate-*l/ 64.5%

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

      11. remove-double-neg 64.5%

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

      12. distribute-frac-neg 64.5%

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

      13. frac-2neg 64.5%

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

      14. distribute-frac-neg 64.5%

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

      15. clear-num 64.5%

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

      16. div-inv 70.0%

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

      17. associate-/r/ 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1.95e63 < u

   1. Initial program 77.7%

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

      1. times-frac 97.4%

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

   3. Simplified 97.4%

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

      1. *-commutative 97.4%

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

      2. clear-num 97.4%

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

      3. frac-2neg 97.4%

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

      4. frac-times 87.0%

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

      5. *-un-lft-identity 87.0%

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

      6. remove-double-neg 87.0%

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

      7. distribute-neg-in 87.0%

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

      8. add-sqr-sqrt 36.5%

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

      9. sqrt-unprod 83.8%

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

      10. sqr-neg 83.8%

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

      11. sqrt-unprod 48.8%

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

      12. add-sqr-sqrt 85.4%

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

      13. sub-neg 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-/r* 88.3%

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

   8. Simplified 88.3%

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

   9. Taylor expanded in t1 around 0 85.3%

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

   10. Taylor expanded in v around 0 77.6%

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

       1. associate-/r* 80.2%

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

       2. associate-*r/ 85.3%

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

       3. associate-*r/ 85.8%

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

   12. Simplified 85.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.00052:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 1.76 \cdot 10^{-37}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq 0.66:\\ \;\;\;\;v \cdot \frac{\frac{t1}{t1 - u}}{t1 + u}\\ \mathbf{elif}\;u \leq 1.95 \cdot 10^{+63}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{-5} \lor \neg \left(u \leq 3.9 \cdot 10^{+64}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.6e-5) (not (<= u 3.9e+64)))
   (* t1 (/ (/ v u) (- t1 u)))
   (/ v (- (* u -2.0) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.6e-5) || !(u <= 3.9e+64)) {
		tmp = t1 * ((v / u) / (t1 - u));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.6d-5)) .or. (.not. (u <= 3.9d+64))) then
        tmp = t1 * ((v / u) / (t1 - u))
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.6e-5) || !(u <= 3.9e+64)) {
		tmp = t1 * ((v / u) / (t1 - u));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.6e-5) or not (u <= 3.9e+64):
		tmp = t1 * ((v / u) / (t1 - u))
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.6e-5) || !(u <= 3.9e+64))
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(t1 - u)));
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.6e-5) || ~((u <= 3.9e+64)))
		tmp = t1 * ((v / u) / (t1 - u));
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.6e-5], N[Not[LessEqual[u, 3.9e+64]], $MachinePrecision]], N[(t1 * N[(N[(v / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.59999999999999992e-5 or 3.8999999999999998e64 < u

   1. Initial program 78.9%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

      1. *-commutative 98.0%

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

      2. clear-num 97.9%

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

      3. frac-2neg 97.9%

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

      4. frac-times 88.0%

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

      5. *-un-lft-identity 88.0%

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

      6. remove-double-neg 88.0%

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

      7. distribute-neg-in 88.0%

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

      8. add-sqr-sqrt 44.7%

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

      9. sqrt-unprod 83.5%

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

      10. sqr-neg 83.5%

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

      11. sqrt-unprod 41.0%

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

      12. add-sqr-sqrt 83.4%

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

      13. sub-neg 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate-/r* 86.2%

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

   8. Simplified 86.2%

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

   9. Taylor expanded in t1 around 0 82.6%

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

   10. Taylor expanded in v around 0 72.9%

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

       1. associate-/r* 77.8%

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

       2. associate-*r/ 82.5%

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

       3. associate-*r/ 83.6%

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

   12. Simplified 83.6%

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

   if -5.59999999999999992e-5 < u < 3.8999999999999998e64

   1. Initial program 69.6%

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

      1. associate-/r* 80.5%

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

      2. *-commutative 80.5%

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

      3. associate-/l* 99.0%

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

      4. associate-/l/ 99.1%

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

      5. +-commutative 99.1%

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

      6. remove-double-neg 99.1%

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

      7. unsub-neg 99.1%

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

      8. div-sub 99.1%

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

      9. sub-neg 99.1%

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

      10. *-inverses 99.1%

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

      11. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. *-commutative 81.6%

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

   7. Simplified 81.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{-5} \lor \neg \left(u \leq 3.9 \cdot 10^{+64}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -0.00044 \lor \neg \left(u \leq 1.15 \cdot 10^{+65}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -0.00044) (not (<= u 1.15e+65)))
   (* (/ v u) (/ (- t1) u))
   (/ v (- (* u -2.0) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -0.00044) || !(u <= 1.15e+65)) {
		tmp = (v / u) * (-t1 / u);
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-0.00044d0)) .or. (.not. (u <= 1.15d+65))) then
        tmp = (v / u) * (-t1 / u)
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -0.00044) || !(u <= 1.15e+65)) {
		tmp = (v / u) * (-t1 / u);
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -0.00044) or not (u <= 1.15e+65):
		tmp = (v / u) * (-t1 / u)
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -0.00044) || !(u <= 1.15e+65))
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -0.00044) || ~((u <= 1.15e+65)))
		tmp = (v / u) * (-t1 / u);
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -0.00044], N[Not[LessEqual[u, 1.15e+65]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.40000000000000016e-4 or 1.15e65 < u

   1. Initial program 78.9%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 80.7%

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

   6. Taylor expanded in t1 around 0 80.0%

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

      1. associate-*r/ 80.0%

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

      2. mul-1-neg 80.0%

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

   8. Simplified 80.0%

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

   if -4.40000000000000016e-4 < u < 1.15e65

   1. Initial program 69.6%

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

      1. associate-/r* 80.5%

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

      2. *-commutative 80.5%

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

      3. associate-/l* 99.0%

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

      4. associate-/l/ 99.1%

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

      5. +-commutative 99.1%

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

      6. remove-double-neg 99.1%

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

      7. unsub-neg 99.1%

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

      8. div-sub 99.1%

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

      9. sub-neg 99.1%

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

      10. *-inverses 99.1%

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

      11. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. *-commutative 81.6%

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

   7. Simplified 81.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.00044 \lor \neg \left(u \leq 1.15 \cdot 10^{+65}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.6 \cdot 10^{+146} \lor \neg \left(u \leq 5 \cdot 10^{+167}\right):\\ \;\;\;\;t1 \cdot \frac{v}{t1 \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7.6e+146) (not (<= u 5e+167)))
   (* t1 (/ v (* t1 u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.6e+146) || !(u <= 5e+167)) {
		tmp = t1 * (v / (t1 * 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 <= (-7.6d+146)) .or. (.not. (u <= 5d+167))) then
        tmp = t1 * (v / (t1 * 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 <= -7.6e+146) || !(u <= 5e+167)) {
		tmp = t1 * (v / (t1 * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -7.6e+146) or not (u <= 5e+167):
		tmp = t1 * (v / (t1 * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -7.6e+146) || !(u <= 5e+167))
		tmp = Float64(t1 * Float64(v / Float64(t1 * u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -7.6e+146) || ~((u <= 5e+167)))
		tmp = t1 * (v / (t1 * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -7.6e+146], N[Not[LessEqual[u, 5e+167]], $MachinePrecision]], N[(t1 * N[(v / N[(t1 * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -7.59999999999999958e146 or 4.9999999999999997e167 < u

   1. Initial program 74.1%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. clear-num 99.1%

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

      3. frac-2neg 99.1%

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

      4. frac-times 91.2%

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

      5. *-un-lft-identity 91.2%

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

      6. remove-double-neg 91.2%

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

      7. distribute-neg-in 91.2%

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

      8. add-sqr-sqrt 46.2%

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

      9. sqrt-unprod 85.3%

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

      10. sqr-neg 85.3%

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

      11. sqrt-unprod 43.5%

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

      12. add-sqr-sqrt 89.7%

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

      13. sub-neg 89.7%

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

   6. Applied egg-rr 89.7%

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

      1. associate-/r* 93.9%

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

   8. Simplified 93.9%

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

   9. Taylor expanded in t1 around 0 92.7%

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

   10. Taylor expanded in v around 0 74.1%

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

       1. associate-/r* 84.9%

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

       2. associate-*r/ 92.6%

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

       3. associate-*r/ 90.1%

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

   12. Simplified 90.1%

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

   13. Taylor expanded in u around 0 50.3%

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

       1. *-commutative 50.3%

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

   15. Simplified 50.3%

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

   if -7.59999999999999958e146 < u < 4.9999999999999997e167

   1. Initial program 74.3%

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

   5. Taylor expanded in t1 around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. neg-mul-1 66.9%

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

   7. Simplified 66.9%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.6 \cdot 10^{+146} \lor \neg \left(u \leq 5 \cdot 10^{+167}\right):\\ \;\;\;\;t1 \cdot \frac{v}{t1 \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+147} \lor \neg \left(u \leq 6 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.55e+147) (not (<= u 6e+171))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.55e+147) || !(u <= 6e+171)) {
		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 <= (-1.55d+147)) .or. (.not. (u <= 6d+171))) 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 <= -1.55e+147) || !(u <= 6e+171)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.55e+147) or not (u <= 6e+171):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.55e+147) || !(u <= 6e+171))
		tmp = Float64(v / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.55e+147) || ~((u <= 6e+171)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.55e+147], N[Not[LessEqual[u, 6e+171]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.55e147 or 6.0000000000000002e171 < u

   1. Initial program 73.7%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. clear-num 99.1%

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

      3. frac-2neg 99.1%

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

      4. frac-times 91.0%

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

      5. *-un-lft-identity 91.0%

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

      6. remove-double-neg 91.0%

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

      7. distribute-neg-in 91.0%

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

      8. add-sqr-sqrt 45.3%

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

      9. sqrt-unprod 85.1%

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

      10. sqr-neg 85.1%

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

      11. sqrt-unprod 44.2%

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

      12. add-sqr-sqrt 89.5%

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

      13. sub-neg 89.5%

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

   6. Applied egg-rr 89.5%

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

      1. associate-/r* 93.9%

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

   8. Simplified 93.9%

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

   9. Taylor expanded in t1 around 0 94.1%

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

   10. Taylor expanded in v around 0 73.7%

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

       1. associate-/r* 86.2%

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

       2. associate-*r/ 93.9%

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

       3. associate-*r/ 89.9%

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

   12. Simplified 89.9%

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

   13. Taylor expanded in t1 around inf 45.2%

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

   if -1.55e147 < u < 6.0000000000000002e171

   1. Initial program 74.4%

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

      1. times-frac 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. Add Preprocessing

   5. Taylor expanded in t1 around inf 67.0%

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

      1. associate-*r/ 67.0%

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

      2. neg-mul-1 67.0%

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

   7. Simplified 67.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+147} \lor \neg \left(u \leq 6 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+109}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 1.9 \cdot 10^{+170}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1e+109) (/ (- v) u) (if (<= u 1.9e+170) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1e+109) {
		tmp = -v / u;
	} else if (u <= 1.9e+170) {
		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 <= (-1d+109)) then
        tmp = -v / u
    else if (u <= 1.9d+170) 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 <= -1e+109) {
		tmp = -v / u;
	} else if (u <= 1.9e+170) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1e+109:
		tmp = -v / u
	elif u <= 1.9e+170:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1e+109)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 1.9e+170)
		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 <= -1e+109)
		tmp = -v / u;
	elseif (u <= 1.9e+170)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1e+109], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 1.9e+170], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -9.99999999999999982e108

   1. Initial program 75.5%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. frac-2neg 99.9%

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

      4. frac-times 90.2%

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

      5. *-un-lft-identity 90.2%

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

      6. remove-double-neg 90.2%

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

      7. distribute-neg-in 90.2%

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

      8. add-sqr-sqrt 51.4%

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

      9. sqrt-unprod 85.4%

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

      10. sqr-neg 85.4%

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

      11. sqrt-unprod 38.8%

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

      12. add-sqr-sqrt 90.2%

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

      13. sub-neg 90.2%

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

   6. Applied egg-rr 90.2%

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

      1. associate-/r* 95.0%

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

   8. Simplified 95.0%

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

   9. Taylor expanded in t1 around inf 37.1%

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

   10. Taylor expanded in t1 around 0 37.1%

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

       1. neg-mul-1 37.1%

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

       2. distribute-neg-frac 37.1%

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

   12. Simplified 37.1%

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

   if -9.99999999999999982e108 < u < 1.8999999999999999e170

   1. Initial program 74.3%

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

      1. times-frac 96.2%

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

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 68.7%

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

      1. associate-*r/ 68.7%

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

      2. neg-mul-1 68.7%

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

   7. Simplified 68.7%

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

   if 1.8999999999999999e170 < u

   1. Initial program 72.2%

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

      1. times-frac 98.4%

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

   3. Simplified 98.4%

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

      1. *-commutative 98.4%

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

      2. clear-num 98.4%

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

      3. frac-2neg 98.4%

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

      4. frac-times 91.2%

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

      5. *-un-lft-identity 91.2%

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

      6. remove-double-neg 91.2%

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

      7. distribute-neg-in 91.2%

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

      8. add-sqr-sqrt 42.4%

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

      9. sqrt-unprod 85.5%

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

      10. sqr-neg 85.5%

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

      11. sqrt-unprod 45.8%

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

      12. add-sqr-sqrt 88.3%

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

      13. sub-neg 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. associate-/r* 91.1%

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

   8. Simplified 91.1%

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

   9. Taylor expanded in t1 around 0 91.3%

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

   10. Taylor expanded in v around 0 72.2%

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

       1. associate-/r* 82.3%

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

       2. associate-*r/ 91.4%

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

       3. associate-*r/ 89.2%

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

   12. Simplified 89.2%

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

   13. Taylor expanded in t1 around inf 51.1%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+109}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 1.9 \cdot 10^{+170}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+109}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+171}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1e+109)
   (/ -0.5 (/ u v))
   (if (<= u 7.2e+171) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1e+109) {
		tmp = -0.5 / (u / v);
	} else if (u <= 7.2e+171) {
		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 <= (-1d+109)) then
        tmp = (-0.5d0) / (u / v)
    else if (u <= 7.2d+171) 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 <= -1e+109) {
		tmp = -0.5 / (u / v);
	} else if (u <= 7.2e+171) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1e+109:
		tmp = -0.5 / (u / v)
	elif u <= 7.2e+171:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1e+109)
		tmp = Float64(-0.5 / Float64(u / v));
	elseif (u <= 7.2e+171)
		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 <= -1e+109)
		tmp = -0.5 / (u / v);
	elseif (u <= 7.2e+171)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1e+109], N[(-0.5 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.2e+171], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -9.99999999999999982e108

   1. Initial program 75.5%

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

      1. associate-/r* 90.0%

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

      2. *-commutative 90.0%

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

      3. associate-/l* 99.8%

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

      4. associate-/l/ 85.5%

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

      5. +-commutative 85.5%

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

      6. remove-double-neg 85.5%

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

      7. unsub-neg 85.5%

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

      8. div-sub 85.5%

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

      9. sub-neg 85.5%

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

      10. *-inverses 85.5%

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

      11. metadata-eval 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in t1 around inf 42.1%

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

      1. mul-1-neg 42.1%

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

      2. unsub-neg 42.1%

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

      3. *-commutative 42.1%

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

   7. Simplified 42.1%

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

   8. Taylor expanded in u around inf 37.1%

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

      1. associate-*r/ 37.1%

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

      2. associate-/l* 37.3%

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

   10. Simplified 37.3%

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

   if -9.99999999999999982e108 < u < 7.20000000000000036e171

   1. Initial program 74.3%

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

      1. times-frac 96.2%

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

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 68.7%

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

      1. associate-*r/ 68.7%

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

      2. neg-mul-1 68.7%

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

   7. Simplified 68.7%

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

   if 7.20000000000000036e171 < u

   1. Initial program 72.2%

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

      1. times-frac 98.4%

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

   3. Simplified 98.4%

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

      1. *-commutative 98.4%

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

      2. clear-num 98.4%

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

      3. frac-2neg 98.4%

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

      4. frac-times 91.2%

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

      5. *-un-lft-identity 91.2%

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

      6. remove-double-neg 91.2%

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

      7. distribute-neg-in 91.2%

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

      8. add-sqr-sqrt 42.4%

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

      9. sqrt-unprod 85.5%

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

      10. sqr-neg 85.5%

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

      11. sqrt-unprod 45.8%

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

      12. add-sqr-sqrt 88.3%

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

      13. sub-neg 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. associate-/r* 91.1%

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

   8. Simplified 91.1%

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

   9. Taylor expanded in t1 around 0 91.3%

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

   10. Taylor expanded in v around 0 72.2%

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

       1. associate-/r* 82.3%

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

       2. associate-*r/ 91.4%

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

       3. associate-*r/ 89.2%

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

   12. Simplified 89.2%

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

   13. Taylor expanded in t1 around inf 51.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+109}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+171}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u \cdot -2 - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- (* u -2.0) t1)))

C

double code(double u, double v, double t1) {
	return v / ((u * -2.0) - 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 * (-2.0d0)) - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Python

def code(u, v, t1):
	return v / ((u * -2.0) - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(u * -2.0) - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((u * -2.0) - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.2%

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

   1. associate-/r* 85.9%

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

   2. *-commutative 85.9%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 93.5%

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

   5. +-commutative 93.5%

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

   6. remove-double-neg 93.5%

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

   7. unsub-neg 93.5%

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

   8. div-sub 93.5%

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

   9. sub-neg 93.5%

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

   10. *-inverses 93.5%

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

   11. metadata-eval 93.5%

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

3. Simplified 93.5%

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

5. Taylor expanded in t1 around inf 65.1%

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

   1. mul-1-neg 65.1%

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

   2. unsub-neg 65.1%

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

   3. *-commutative 65.1%

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

7. Simplified 65.1%

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

8. Final simplification 65.1%

   \[\leadsto \frac{v}{u \cdot -2 - t1} \]
9. Add Preprocessing

Alternative 14: 17.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return v / 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 / u
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. times-frac 97.0%

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

3. Simplified 97.0%

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

   1. *-commutative 97.0%

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

   2. clear-num 96.2%

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

   3. frac-2neg 96.2%

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

   4. frac-times 82.9%

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

   5. *-un-lft-identity 82.9%

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

   6. remove-double-neg 82.9%

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

   7. distribute-neg-in 82.9%

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

   8. add-sqr-sqrt 40.4%

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

   9. sqrt-unprod 64.9%

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

   10. sqr-neg 64.9%

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

   11. sqrt-unprod 26.8%

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

   12. add-sqr-sqrt 54.9%

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

   13. sub-neg 54.9%

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

6. Applied egg-rr 54.9%

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

   1. associate-/r* 58.5%

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

8. Simplified 58.5%

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

9. Taylor expanded in t1 around 0 50.9%

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

10. Taylor expanded in v around 0 45.8%

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

    1. associate-/r* 49.4%

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

    2. associate-*r/ 50.1%

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

    3. associate-*r/ 50.0%

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

12. Simplified 50.0%

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

13. Taylor expanded in t1 around inf 17.7%

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

14. Final simplification 17.7%

    \[\leadsto \frac{v}{u} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))