Rosa's DopplerBench

Percentage Accurate: 73.4% → 97.8%

Time: 10.3s

Alternatives: 16

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: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 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. Final simplification 98.4%

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

Alternative 2: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1 + u}{v}\\ \mathbf{if}\;u \leq -5.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{t1}{t1 - u}}{t_1}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t_1}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (+ t1 u) v)))
   (if (<= u -5.8e-91)
     (/ (/ t1 (- t1 u)) t_1)
     (if (<= u 5.2e-14) (/ (- v) t1) (/ (/ t1 t_1) (- t1 u))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 + u) / v;
	double tmp;
	if (u <= -5.8e-91) {
		tmp = (t1 / (t1 - u)) / t_1;
	} else if (u <= 5.2e-14) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / t_1) / (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 = (t1 + u) / v
    if (u <= (-5.8d-91)) then
        tmp = (t1 / (t1 - u)) / t_1
    else if (u <= 5.2d-14) then
        tmp = -v / t1
    else
        tmp = (t1 / t_1) / (t1 - u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 + u) / v;
	double tmp;
	if (u <= -5.8e-91) {
		tmp = (t1 / (t1 - u)) / t_1;
	} else if (u <= 5.2e-14) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / t_1) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 + u) / v
	tmp = 0
	if u <= -5.8e-91:
		tmp = (t1 / (t1 - u)) / t_1
	elif u <= 5.2e-14:
		tmp = -v / t1
	else:
		tmp = (t1 / t_1) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 + u) / v)
	tmp = 0.0
	if (u <= -5.8e-91)
		tmp = Float64(Float64(t1 / Float64(t1 - u)) / t_1);
	elseif (u <= 5.2e-14)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 / t_1) / Float64(t1 - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 + u) / v;
	tmp = 0.0;
	if (u <= -5.8e-91)
		tmp = (t1 / (t1 - u)) / t_1;
	elseif (u <= 5.2e-14)
		tmp = -v / t1;
	else
		tmp = (t1 / t_1) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]}, If[LessEqual[u, -5.8e-91], N[(N[(t1 / N[(t1 - u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[u, 5.2e-14], N[((-v) / t1), $MachinePrecision], N[(N[(t1 / t$95$1), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.8000000000000001e-91

   1. Initial program 80.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 98.7%

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

      3. frac-2neg 98.7%

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

      4. frac-times 93.0%

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

      5. *-un-lft-identity 93.0%

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

      6. remove-double-neg 93.0%

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

      7. distribute-neg-in 93.0%

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

      8. add-sqr-sqrt 47.0%

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

      9. sqrt-unprod 83.0%

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

      10. sqr-neg 83.0%

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

      11. sqrt-unprod 42.0%

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

      12. add-sqr-sqrt 79.2%

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

      13. sub-neg 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/r* 81.3%

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

   8. Simplified 81.3%

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

   if -5.8000000000000001e-91 < u < 5.19999999999999993e-14

   1. Initial program 66.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around inf 88.3%

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

      1. associate-*r/ 88.3%

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

      2. neg-mul-1 88.3%

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

   7. Simplified 88.3%

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

   if 5.19999999999999993e-14 < u

   1. Initial program 80.2%

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

      1. times-frac 94.8%

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

   3. Simplified 94.8%

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

      1. *-commutative 94.8%

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

      2. clear-num 94.7%

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

      3. frac-2neg 94.7%

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

      4. frac-times 89.2%

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

      5. *-un-lft-identity 89.2%

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

      6. remove-double-neg 89.2%

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

      7. distribute-neg-in 89.2%

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

      8. add-sqr-sqrt 51.9%

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

      9. sqrt-unprod 86.4%

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

      10. sqr-neg 86.4%

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

      11. sqrt-unprod 36.0%

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

      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* 88.8%

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

   8. Simplified 88.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{t1}{t1 - u}}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 + u}{v}}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{-42} \lor \neg \left(u \leq 5.2 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.3e-42) (not (<= u 5.2e-14)))
   (/ t1 (* (- t1 u) (/ u v)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e-42) || !(u <= 5.2e-14)) {
		tmp = t1 / ((t1 - u) * (u / v));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.3d-42)) .or. (.not. (u <= 5.2d-14))) then
        tmp = t1 / ((t1 - u) * (u / v))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e-42) || !(u <= 5.2e-14)) {
		tmp = t1 / ((t1 - u) * (u / v));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.3e-42) or not (u <= 5.2e-14):
		tmp = t1 / ((t1 - u) * (u / v))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.3e-42) || !(u <= 5.2e-14))
		tmp = Float64(t1 / Float64(Float64(t1 - u) * Float64(u / v)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.3e-42) || ~((u <= 5.2e-14)))
		tmp = t1 / ((t1 - u) * (u / v));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.3e-42], N[Not[LessEqual[u, 5.2e-14]], $MachinePrecision]], N[(t1 / N[(N[(t1 - u), $MachinePrecision] * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.30000000000000004e-42 or 5.19999999999999993e-14 < u

   1. Initial program 81.3%

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

      1. times-frac 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t1 around 0 80.1%

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

      1. *-commutative 80.1%

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

      2. clear-num 80.0%

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

      3. frac-2neg 80.0%

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

      4. frac-times 83.1%

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

      5. *-un-lft-identity 83.1%

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

      6. remove-double-neg 83.1%

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

      7. distribute-neg-in 83.1%

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

      8. add-sqr-sqrt 43.2%

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

      9. sqrt-unprod 83.1%

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

      10. sqr-neg 83.1%

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

      11. sqrt-unprod 40.0%

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

      12. add-sqr-sqrt 83.1%

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

      13. sub-neg 83.1%

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

   7. Applied egg-rr 83.1%

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

   if -2.30000000000000004e-42 < u < 5.19999999999999993e-14

   1. Initial program 65.8%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 85.8%

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

      1. associate-*r/ 85.8%

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

      2. neg-mul-1 85.8%

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

   7. Simplified 85.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{-42} \lor \neg \left(u \leq 5.2 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-42}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{-12}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.7e-42)
   (/ t1 (* (- t1 u) (/ u v)))
   (if (<= u 1.35e-12) (/ (- v) t1) (/ (* t1 (/ v u)) (- t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.7e-42) {
		tmp = t1 / ((t1 - u) * (u / v));
	} else if (u <= 1.35e-12) {
		tmp = -v / t1;
	} 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) :: tmp
    if (u <= (-1.7d-42)) then
        tmp = t1 / ((t1 - u) * (u / v))
    else if (u <= 1.35d-12) then
        tmp = -v / t1
    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 tmp;
	if (u <= -1.7e-42) {
		tmp = t1 / ((t1 - u) * (u / v));
	} else if (u <= 1.35e-12) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / u)) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.7e-42:
		tmp = t1 / ((t1 - u) * (u / v))
	elif u <= 1.35e-12:
		tmp = -v / t1
	else:
		tmp = (t1 * (v / u)) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.7e-42)
		tmp = Float64(t1 / Float64(Float64(t1 - u) * Float64(u / v)));
	elseif (u <= 1.35e-12)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.7e-42)
		tmp = t1 / ((t1 - u) * (u / v));
	elseif (u <= 1.35e-12)
		tmp = -v / t1;
	else
		tmp = (t1 * (v / u)) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.7e-42], N[(t1 / N[(N[(t1 - u), $MachinePrecision] * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.35e-12], N[((-v) / t1), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.70000000000000011e-42

   1. Initial program 82.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. clear-num 77.9%

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

      3. frac-2neg 77.9%

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

      4. frac-times 82.9%

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

      5. *-un-lft-identity 82.9%

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

      6. remove-double-neg 82.9%

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

      7. distribute-neg-in 82.9%

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

      8. add-sqr-sqrt 39.6%

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

      9. sqrt-unprod 82.8%

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

      10. sqr-neg 82.8%

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

      11. sqrt-unprod 43.4%

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

      12. add-sqr-sqrt 82.6%

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

      13. sub-neg 82.6%

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

   7. Applied egg-rr 82.6%

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

   if -1.70000000000000011e-42 < u < 1.3499999999999999e-12

   1. Initial program 65.8%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 85.8%

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

      1. associate-*r/ 85.8%

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

      2. neg-mul-1 85.8%

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

   7. Simplified 85.8%

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

   if 1.3499999999999999e-12 < u

   1. Initial program 80.2%

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

      1. times-frac 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in t1 around 0 82.3%

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

      1. associate-*l/ 87.3%

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

      2. frac-2neg 87.3%

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

      3. add-sqr-sqrt 47.1%

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

      4. sqrt-unprod 54.2%

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

      5. sqr-neg 54.2%

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

      6. sqrt-unprod 19.3%

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

      7. add-sqr-sqrt 47.9%

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

      8. distribute-lft-neg-out 47.9%

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

      9. add-sqr-sqrt 28.6%

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

      10. sqrt-unprod 55.9%

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

      11. sqr-neg 55.9%

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

      12. sqrt-unprod 40.0%

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

      13. add-sqr-sqrt 87.3%

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

      14. distribute-neg-in 87.3%

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

      15. add-sqr-sqrt 47.2%

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

      16. sqrt-unprod 85.0%

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

      17. sqr-neg 85.0%

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

      18. sqrt-unprod 40.1%

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

      19. add-sqr-sqrt 87.6%

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

      20. sub-neg 87.6%

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

   7. Applied egg-rr 87.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-42}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{-12}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{-90}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.15e-90)
   (* (/ v (+ t1 u)) (/ (- t1) u))
   (if (<= u 3.1e-9) (/ (- v) t1) (/ (* t1 (/ v u)) (- t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e-90) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else if (u <= 3.1e-9) {
		tmp = -v / t1;
	} 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) :: tmp
    if (u <= (-1.15d-90)) then
        tmp = (v / (t1 + u)) * (-t1 / u)
    else if (u <= 3.1d-9) then
        tmp = -v / t1
    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 tmp;
	if (u <= -1.15e-90) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else if (u <= 3.1e-9) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / u)) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.15e-90:
		tmp = (v / (t1 + u)) * (-t1 / u)
	elif u <= 3.1e-9:
		tmp = -v / t1
	else:
		tmp = (t1 * (v / u)) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.15e-90)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	elseif (u <= 3.1e-9)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.15e-90)
		tmp = (v / (t1 + u)) * (-t1 / u);
	elseif (u <= 3.1e-9)
		tmp = -v / t1;
	else
		tmp = (t1 * (v / u)) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.15e-90], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.1e-9], N[((-v) / t1), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.1499999999999999e-90

   1. Initial program 80.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around 0 80.9%

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

      1. associate-*r/ 76.1%

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

      2. mul-1-neg 76.1%

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

   7. Simplified 80.9%

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

   if -1.1499999999999999e-90 < u < 3.10000000000000005e-9

   1. Initial program 66.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around inf 88.3%

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

      1. associate-*r/ 88.3%

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

      2. neg-mul-1 88.3%

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

   7. Simplified 88.3%

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

   if 3.10000000000000005e-9 < u

   1. Initial program 80.2%

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

      1. times-frac 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in t1 around 0 82.3%

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

      1. associate-*l/ 87.3%

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

      2. frac-2neg 87.3%

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

      3. add-sqr-sqrt 47.1%

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

      4. sqrt-unprod 54.2%

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

      5. sqr-neg 54.2%

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

      6. sqrt-unprod 19.3%

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

      7. add-sqr-sqrt 47.9%

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

      8. distribute-lft-neg-out 47.9%

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

      9. add-sqr-sqrt 28.6%

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

      10. sqrt-unprod 55.9%

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

      11. sqr-neg 55.9%

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

      12. sqrt-unprod 40.0%

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

      13. add-sqr-sqrt 87.3%

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

      14. distribute-neg-in 87.3%

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

      15. add-sqr-sqrt 47.2%

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

      16. sqrt-unprod 85.0%

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

      17. sqr-neg 85.0%

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

      18. sqrt-unprod 40.1%

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

      19. add-sqr-sqrt 87.6%

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

      20. sub-neg 87.6%

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

   7. Applied egg-rr 87.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{-90}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{t1}{t1 - u}}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;u \leq 1.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.1e-90)
   (/ (/ t1 (- t1 u)) (/ (+ t1 u) v))
   (if (<= u 1.5e-13) (/ (- v) t1) (/ (* t1 (/ v u)) (- t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e-90) {
		tmp = (t1 / (t1 - u)) / ((t1 + u) / v);
	} else if (u <= 1.5e-13) {
		tmp = -v / t1;
	} 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) :: tmp
    if (u <= (-2.1d-90)) then
        tmp = (t1 / (t1 - u)) / ((t1 + u) / v)
    else if (u <= 1.5d-13) then
        tmp = -v / t1
    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 tmp;
	if (u <= -2.1e-90) {
		tmp = (t1 / (t1 - u)) / ((t1 + u) / v);
	} else if (u <= 1.5e-13) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / u)) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.1e-90:
		tmp = (t1 / (t1 - u)) / ((t1 + u) / v)
	elif u <= 1.5e-13:
		tmp = -v / t1
	else:
		tmp = (t1 * (v / u)) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.1e-90)
		tmp = Float64(Float64(t1 / Float64(t1 - u)) / Float64(Float64(t1 + u) / v));
	elseif (u <= 1.5e-13)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.1e-90)
		tmp = (t1 / (t1 - u)) / ((t1 + u) / v);
	elseif (u <= 1.5e-13)
		tmp = -v / t1;
	else
		tmp = (t1 * (v / u)) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.1e-90], N[(N[(t1 / N[(t1 - u), $MachinePrecision]), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.5e-13], N[((-v) / t1), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.0999999999999999e-90

   1. Initial program 80.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 98.7%

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

      3. frac-2neg 98.7%

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

      4. frac-times 93.0%

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

      5. *-un-lft-identity 93.0%

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

      6. remove-double-neg 93.0%

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

      7. distribute-neg-in 93.0%

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

      8. add-sqr-sqrt 47.0%

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

      9. sqrt-unprod 83.0%

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

      10. sqr-neg 83.0%

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

      11. sqrt-unprod 42.0%

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

      12. add-sqr-sqrt 79.2%

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

      13. sub-neg 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/r* 81.3%

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

   8. Simplified 81.3%

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

   if -2.0999999999999999e-90 < u < 1.49999999999999992e-13

   1. Initial program 66.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around inf 88.3%

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

      1. associate-*r/ 88.3%

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

      2. neg-mul-1 88.3%

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

   7. Simplified 88.3%

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

   if 1.49999999999999992e-13 < u

   1. Initial program 80.2%

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

      1. times-frac 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in t1 around 0 82.3%

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

      1. associate-*l/ 87.3%

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

      2. frac-2neg 87.3%

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

      3. add-sqr-sqrt 47.1%

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

      4. sqrt-unprod 54.2%

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

      5. sqr-neg 54.2%

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

      6. sqrt-unprod 19.3%

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

      7. add-sqr-sqrt 47.9%

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

      8. distribute-lft-neg-out 47.9%

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

      9. add-sqr-sqrt 28.6%

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

      10. sqrt-unprod 55.9%

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

      11. sqr-neg 55.9%

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

      12. sqrt-unprod 40.0%

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

      13. add-sqr-sqrt 87.3%

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

      14. distribute-neg-in 87.3%

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

      15. add-sqr-sqrt 47.2%

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

      16. sqrt-unprod 85.0%

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

      17. sqr-neg 85.0%

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

      18. sqrt-unprod 40.1%

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

      19. add-sqr-sqrt 87.6%

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

      20. sub-neg 87.6%

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

   7. Applied egg-rr 87.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{t1}{t1 - u}}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;u \leq 1.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{-90} \lor \neg \left(u \leq 8 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.1e-90) (not (<= u 8e-8)))
   (* (/ v u) (/ (- t1) u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.1e-90) || !(u <= 8e-8)) {
		tmp = (v / u) * (-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 <= (-2.1d-90)) .or. (.not. (u <= 8d-8))) then
        tmp = (v / u) * (-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 <= -2.1e-90) || !(u <= 8e-8)) {
		tmp = (v / u) * (-t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.1e-90) or not (u <= 8e-8):
		tmp = (v / u) * (-t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.1e-90) || !(u <= 8e-8))
		tmp = Float64(Float64(v / u) * Float64(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 <= -2.1e-90) || ~((u <= 8e-8)))
		tmp = (v / u) * (-t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.1e-90], N[Not[LessEqual[u, 8e-8]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.0999999999999999e-90 or 8.0000000000000002e-8 < u

   1. Initial program 80.2%

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

      1. times-frac 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in t1 around 0 79.0%

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

   6. Taylor expanded in t1 around 0 78.9%

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

      1. associate-*r/ 78.9%

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

      2. mul-1-neg 78.9%

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

   8. Simplified 78.9%

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

   if -2.0999999999999999e-90 < u < 8.0000000000000002e-8

   1. Initial program 66.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around inf 88.3%

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

      1. associate-*r/ 88.3%

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

      2. neg-mul-1 88.3%

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

   7. Simplified 88.3%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{-90} \lor \neg \left(u \leq 8 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+130} \lor \neg \left(u \leq 3.4 \cdot 10^{+198}\right):\\ \;\;\;\;\frac{t1}{\frac{t1 \cdot u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.5e+130) (not (<= u 3.4e+198)))
   (/ t1 (/ (* t1 u) v))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.5e+130) || !(u <= 3.4e+198)) {
		tmp = t1 / ((t1 * u) / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-6.5d+130)) .or. (.not. (u <= 3.4d+198))) then
        tmp = t1 / ((t1 * u) / v)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.5e+130) || !(u <= 3.4e+198)) {
		tmp = t1 / ((t1 * u) / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.5e+130) or not (u <= 3.4e+198):
		tmp = t1 / ((t1 * u) / v)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.5e+130) || !(u <= 3.4e+198))
		tmp = Float64(t1 / Float64(Float64(t1 * u) / v));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.5e+130) || ~((u <= 3.4e+198)))
		tmp = t1 / ((t1 * u) / v);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.5e+130], N[Not[LessEqual[u, 3.4e+198]], $MachinePrecision]], N[(t1 / N[(N[(t1 * u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.5e130 or 3.4e198 < u

   1. Initial program 79.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around 0 92.5%

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

      1. *-commutative 92.5%

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

      2. clear-num 92.4%

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

      3. frac-2neg 92.4%

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

      4. frac-times 86.6%

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

      5. *-un-lft-identity 86.6%

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

      6. remove-double-neg 86.6%

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

      7. distribute-neg-in 86.6%

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

      8. add-sqr-sqrt 41.7%

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

      9. sqrt-unprod 86.5%

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

      10. sqr-neg 86.5%

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

      11. sqrt-unprod 44.9%

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

      12. add-sqr-sqrt 86.4%

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

      13. sub-neg 86.4%

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

   7. Applied egg-rr 86.4%

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

   8. Taylor expanded in u around 0 40.7%

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

   if -6.5e130 < u < 3.4e198

   1. Initial program 72.4%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around inf 62.7%

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

      1. associate-*r/ 62.7%

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

      2. neg-mul-1 62.7%

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

   7. Simplified 62.7%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+130} \lor \neg \left(u \leq 3.4 \cdot 10^{+198}\right):\\ \;\;\;\;\frac{t1}{\frac{t1 \cdot u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+138}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+198}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot -0.5}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.6e+138)
   (/ v u)
   (if (<= u 3.4e+198) (/ (- v) t1) (/ (* v -0.5) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.6e+138) {
		tmp = v / u;
	} else if (u <= 3.4e+198) {
		tmp = -v / t1;
	} else {
		tmp = (v * -0.5) / 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 <= (-5.6d+138)) then
        tmp = v / u
    else if (u <= 3.4d+198) then
        tmp = -v / t1
    else
        tmp = (v * (-0.5d0)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.6e+138) {
		tmp = v / u;
	} else if (u <= 3.4e+198) {
		tmp = -v / t1;
	} else {
		tmp = (v * -0.5) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.6e+138:
		tmp = v / u
	elif u <= 3.4e+198:
		tmp = -v / t1
	else:
		tmp = (v * -0.5) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.6e+138)
		tmp = Float64(v / u);
	elseif (u <= 3.4e+198)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v * -0.5) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.6e+138)
		tmp = v / u;
	elseif (u <= 3.4e+198)
		tmp = -v / t1;
	else
		tmp = (v * -0.5) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.6e+138], N[(v / u), $MachinePrecision], If[LessEqual[u, 3.4e+198], N[((-v) / t1), $MachinePrecision], N[(N[(v * -0.5), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.6000000000000002e138

   1. Initial program 80.7%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 91.9%

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

      1. associate-*l/ 91.7%

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

      2. frac-2neg 91.7%

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

      3. add-sqr-sqrt 40.0%

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

      4. sqrt-unprod 80.4%

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

      5. sqr-neg 80.4%

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

      6. sqrt-unprod 46.4%

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

      7. add-sqr-sqrt 81.2%

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

      8. distribute-lft-neg-out 81.2%

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

      9. add-sqr-sqrt 34.8%

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

      10. sqrt-unprod 77.7%

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

      11. sqr-neg 77.7%

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

      12. sqrt-unprod 51.6%

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

      13. add-sqr-sqrt 91.7%

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

      14. distribute-neg-in 91.7%

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

      15. add-sqr-sqrt 40.1%

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

      16. sqrt-unprod 89.1%

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

      17. sqr-neg 89.1%

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

      18. sqrt-unprod 51.9%

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

      19. add-sqr-sqrt 91.9%

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

      20. sub-neg 91.9%

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in t1 around inf 32.0%

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

   if -5.6000000000000002e138 < u < 3.4e198

   1. Initial program 72.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 inf 62.6%

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

      1. associate-*r/ 62.6%

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

      2. neg-mul-1 62.6%

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

   7. Simplified 62.6%

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

   if 3.4e198 < u

   1. Initial program 76.4%

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

   3. Taylor expanded in t1 around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. unpow2 71.6%

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

      3. associate-*r* 71.6%

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

      4. distribute-rgt-out 76.4%

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

   5. Simplified 76.4%

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

   6. Taylor expanded in t1 around inf 32.4%

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

      1. associate-*r/ 32.4%

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

   8. Simplified 32.4%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot v}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+138}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+198}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot -0.5}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.6 \cdot 10^{+130}:\\ \;\;\;\;\frac{t1}{t1 \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 10^{+199}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot -0.5}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6.6e+130)
   (/ t1 (* t1 (/ u v)))
   (if (<= u 1e+199) (/ (- v) t1) (/ (* v -0.5) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.6e+130) {
		tmp = t1 / (t1 * (u / v));
	} else if (u <= 1e+199) {
		tmp = -v / t1;
	} else {
		tmp = (v * -0.5) / 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 <= (-6.6d+130)) then
        tmp = t1 / (t1 * (u / v))
    else if (u <= 1d+199) then
        tmp = -v / t1
    else
        tmp = (v * (-0.5d0)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.6e+130) {
		tmp = t1 / (t1 * (u / v));
	} else if (u <= 1e+199) {
		tmp = -v / t1;
	} else {
		tmp = (v * -0.5) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6.6e+130:
		tmp = t1 / (t1 * (u / v))
	elif u <= 1e+199:
		tmp = -v / t1
	else:
		tmp = (v * -0.5) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6.6e+130)
		tmp = Float64(t1 / Float64(t1 * Float64(u / v)));
	elseif (u <= 1e+199)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v * -0.5) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6.6e+130)
		tmp = t1 / (t1 * (u / v));
	elseif (u <= 1e+199)
		tmp = -v / t1;
	else
		tmp = (v * -0.5) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6.6e+130], N[(t1 / N[(t1 * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1e+199], N[((-v) / t1), $MachinePrecision], N[(N[(v * -0.5), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -6.6e130

   1. Initial program 81.7%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 89.5%

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

      1. *-commutative 89.5%

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

      2. clear-num 89.4%

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

      3. frac-2neg 89.4%

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

      4. frac-times 89.5%

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

      5. *-un-lft-identity 89.5%

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

      6. remove-double-neg 89.5%

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

      7. distribute-neg-in 89.5%

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

      8. add-sqr-sqrt 40.7%

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

      9. sqrt-unprod 89.3%

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

      10. sqr-neg 89.3%

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

      11. sqrt-unprod 48.8%

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

      12. add-sqr-sqrt 89.3%

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

      13. sub-neg 89.3%

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

   7. Applied egg-rr 89.3%

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

   8. Taylor expanded in u around 0 43.5%

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

      1. associate-*r/ 35.7%

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

   10. Simplified 35.7%

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

   if -6.6e130 < u < 1.0000000000000001e199

   1. Initial program 72.4%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around inf 62.7%

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

      1. associate-*r/ 62.7%

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

      2. neg-mul-1 62.7%

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

   7. Simplified 62.7%

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

   if 1.0000000000000001e199 < u

   1. Initial program 76.4%

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

   3. Taylor expanded in t1 around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. unpow2 71.6%

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

      3. associate-*r* 71.6%

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

      4. distribute-rgt-out 76.4%

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

   5. Simplified 76.4%

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

   6. Taylor expanded in t1 around inf 32.4%

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

      1. associate-*r/ 32.4%

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

   8. Simplified 32.4%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot v}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.6 \cdot 10^{+130}:\\ \;\;\;\;\frac{t1}{t1 \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 10^{+199}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot -0.5}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.85 \cdot 10^{+139} \lor \neg \left(u \leq 6.6 \cdot 10^{+198}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.85e+139) (not (<= u 6.6e+198))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.85e+139) || !(u <= 6.6e+198)) {
		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 <= (-1.85d+139)) .or. (.not. (u <= 6.6d+198))) 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 <= -1.85e+139) || !(u <= 6.6e+198)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.85e+139) or not (u <= 6.6e+198):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.85e+139) || !(u <= 6.6e+198))
		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 <= -1.85e+139) || ~((u <= 6.6e+198)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.85e+139], N[Not[LessEqual[u, 6.6e+198]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.84999999999999996e139 or 6.59999999999999988e198 < u

   1. Initial program 79.1%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 94.1%

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

      1. associate-*l/ 94.0%

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

      2. frac-2neg 94.0%

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

      3. add-sqr-sqrt 42.8%

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

      4. sqrt-unprod 80.3%

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

      5. sqr-neg 80.3%

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

      6. sqrt-unprod 43.1%

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

      7. add-sqr-sqrt 79.4%

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

      8. distribute-lft-neg-out 79.4%

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

      9. add-sqr-sqrt 36.3%

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

      10. sqrt-unprod 75.8%

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

      11. sqr-neg 75.8%

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

      12. sqrt-unprod 51.1%

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

      13. add-sqr-sqrt 94.0%

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

      14. distribute-neg-in 94.0%

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

      15. add-sqr-sqrt 42.9%

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

      16. sqrt-unprod 88.0%

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

      17. sqr-neg 88.0%

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

      18. sqrt-unprod 51.2%

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

      19. add-sqr-sqrt 94.0%

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

      20. sub-neg 94.0%

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

   7. Applied egg-rr 94.0%

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

   8. Taylor expanded in t1 around inf 31.8%

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

   if -1.84999999999999996e139 < u < 6.59999999999999988e198

   1. Initial program 72.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 inf 62.6%

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

      1. associate-*r/ 62.6%

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

      2. neg-mul-1 62.6%

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

   7. Simplified 62.6%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.85 \cdot 10^{+139} \lor \neg \left(u \leq 6.6 \cdot 10^{+198}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.35 \cdot 10^{+138}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 3.9 \cdot 10^{+198}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.35e+138) (/ v u) (if (<= u 3.9e+198) (/ (- v) t1) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.35e+138) {
		tmp = v / u;
	} else if (u <= 3.9e+198) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.35d+138)) then
        tmp = v / u
    else if (u <= 3.9d+198) then
        tmp = -v / t1
    else
        tmp = -v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.35e+138) {
		tmp = v / u;
	} else if (u <= 3.9e+198) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.35e+138:
		tmp = v / u
	elif u <= 3.9e+198:
		tmp = -v / t1
	else:
		tmp = -v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.35e+138)
		tmp = Float64(v / u);
	elseif (u <= 3.9e+198)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-v) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.35e+138)
		tmp = v / u;
	elseif (u <= 3.9e+198)
		tmp = -v / t1;
	else
		tmp = -v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.35e+138], N[(v / u), $MachinePrecision], If[LessEqual[u, 3.9e+198], N[((-v) / t1), $MachinePrecision], N[((-v) / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.3499999999999999e138

   1. Initial program 80.7%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 91.9%

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

      1. associate-*l/ 91.7%

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

      2. frac-2neg 91.7%

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

      3. add-sqr-sqrt 40.0%

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

      4. sqrt-unprod 80.4%

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

      5. sqr-neg 80.4%

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

      6. sqrt-unprod 46.4%

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

      7. add-sqr-sqrt 81.2%

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

      8. distribute-lft-neg-out 81.2%

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

      9. add-sqr-sqrt 34.8%

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

      10. sqrt-unprod 77.7%

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

      11. sqr-neg 77.7%

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

      12. sqrt-unprod 51.6%

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

      13. add-sqr-sqrt 91.7%

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

      14. distribute-neg-in 91.7%

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

      15. add-sqr-sqrt 40.1%

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

      16. sqrt-unprod 89.1%

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

      17. sqr-neg 89.1%

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

      18. sqrt-unprod 51.9%

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

      19. add-sqr-sqrt 91.9%

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

      20. sub-neg 91.9%

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in t1 around inf 32.0%

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

   if -2.3499999999999999e138 < u < 3.9e198

   1. Initial program 72.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 inf 62.6%

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

      1. associate-*r/ 62.6%

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

      2. neg-mul-1 62.6%

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

   7. Simplified 62.6%

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

   if 3.9e198 < u

   1. Initial program 76.4%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 97.8%

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

   6. Taylor expanded in t1 around inf 32.4%

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

      1. associate-*r/ 32.4%

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

      2. neg-mul-1 32.4%

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

   8. Simplified 32.4%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.35 \cdot 10^{+138}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 3.9 \cdot 10^{+198}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 22.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.6 \cdot 10^{+87} \lor \neg \left(t1 \leq 1.35 \cdot 10^{+179}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.6e+87) (not (<= t1 1.35e+179))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.6e+87) || !(t1 <= 1.35e+179)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4.6d+87)) .or. (.not. (t1 <= 1.35d+179))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.6e+87) || !(t1 <= 1.35e+179)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.6e+87) or not (t1 <= 1.35e+179):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.6e+87) || !(t1 <= 1.35e+179))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.6e+87) || ~((t1 <= 1.35e+179)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.6e+87], N[Not[LessEqual[t1, 1.35e+179]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.6000000000000003e87 or 1.34999999999999991e179 < t1

   1. Initial program 40.4%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
   4. 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 99.2%

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

      3. frac-2neg 99.2%

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

      4. frac-times 68.2%

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

      5. *-un-lft-identity 68.2%

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

      6. remove-double-neg 68.2%

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

      7. distribute-neg-in 68.2%

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

      8. add-sqr-sqrt 45.7%

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

      9. sqrt-unprod 41.8%

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

      10. sqr-neg 41.8%

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

      11. sqrt-unprod 12.6%

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

      12. add-sqr-sqrt 31.2%

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

      13. sub-neg 31.2%

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

   6. Applied egg-rr 31.2%

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

      1. associate-/r* 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in t1 around inf 29.0%

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

   if -4.6000000000000003e87 < t1 < 1.34999999999999991e179

   1. Initial program 84.0%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around 0 68.5%

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

      1. associate-*l/ 69.4%

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

      2. frac-2neg 69.4%

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

      3. add-sqr-sqrt 34.6%

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

      4. sqrt-unprod 41.5%

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

      5. sqr-neg 41.5%

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

      6. sqrt-unprod 17.6%

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

      7. add-sqr-sqrt 36.1%

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

      8. distribute-lft-neg-out 36.1%

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

      9. add-sqr-sqrt 18.6%

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

      10. sqrt-unprod 41.1%

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

      11. sqr-neg 41.1%

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

      12. sqrt-unprod 34.7%

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

      13. add-sqr-sqrt 69.4%

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

      14. distribute-neg-in 69.4%

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

      15. add-sqr-sqrt 34.7%

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

      16. sqrt-unprod 70.0%

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

      17. sqr-neg 70.0%

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

      18. sqrt-unprod 33.8%

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

      19. add-sqr-sqrt 68.5%

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

      20. sub-neg 68.5%

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

   7. Applied egg-rr 68.5%

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

   8. Taylor expanded in t1 around inf 16.0%

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

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.6 \cdot 10^{+87} \lor \neg \left(t1 \leq 1.35 \cdot 10^{+179}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. associate-/r* 83.3%

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

   2. *-commutative 83.3%

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

   3. associate-/l* 97.5%

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

   4. associate-/l/ 91.5%

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

   5. +-commutative 91.5%

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

   6. remove-double-neg 91.5%

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

   7. unsub-neg 91.5%

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

   8. div-sub 91.5%

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

   9. sub-neg 91.5%

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

   10. *-inverses 91.5%

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

   11. metadata-eval 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in v around 0 91.5%

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

   1. associate-/r* 98.2%

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

   2. fma-neg 98.2%

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

   3. metadata-eval 98.2%

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

7. Simplified 98.2%

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

   1. frac-2neg 98.2%

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

   2. distribute-frac-neg 98.2%

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

   3. fma-udef 98.2%

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

   4. distribute-neg-in 98.2%

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

   5. mul-1-neg 98.2%

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

   6. add-sqr-sqrt 48.9%

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

   7. sqrt-unprod 74.7%

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

   8. sqr-neg 74.7%

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

   9. sqrt-unprod 36.6%

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

   10. add-sqr-sqrt 72.2%

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

   11. distribute-frac-neg 72.2%

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

   12. frac-2neg 72.2%

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

   13. add-sqr-sqrt 35.6%

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

   14. sqrt-unprod 74.6%

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

   15. sqr-neg 74.6%

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

   16. sqrt-unprod 49.2%

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

   17. add-sqr-sqrt 98.2%

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

   18. distribute-frac-neg 98.2%

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

   19. frac-2neg 98.2%

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

   20. metadata-eval 98.2%

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

9. Applied egg-rr 98.2%

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

10. Final simplification 98.2%

    \[\leadsto \frac{\frac{-v}{t1 + u}}{\frac{u}{t1} + 1} \]
11. Add Preprocessing

Alternative 15: 62.3% 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 74.1%

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

   1. associate-/r* 83.3%

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

   2. *-commutative 83.3%

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

   3. associate-/l* 97.5%

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

   4. associate-/l/ 91.5%

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

   5. +-commutative 91.5%

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

   6. remove-double-neg 91.5%

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

   7. unsub-neg 91.5%

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

   8. div-sub 91.5%

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

   9. sub-neg 91.5%

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

   10. *-inverses 91.5%

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

   11. metadata-eval 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in t1 around inf 56.8%

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

   1. mul-1-neg 56.8%

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

   2. unsub-neg 56.8%

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

   3. *-commutative 56.8%

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

7. Simplified 56.8%

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

8. Final simplification 56.8%

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

Alternative 16: 14.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 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 98.1%

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

   3. frac-2neg 98.1%

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

   4. frac-times 86.7%

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

   5. *-un-lft-identity 86.7%

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

   6. remove-double-neg 86.7%

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

   7. distribute-neg-in 86.7%

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

   8. add-sqr-sqrt 43.5%

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

   9. sqrt-unprod 67.0%

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

   10. sqr-neg 67.0%

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

   11. sqrt-unprod 27.2%

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

   12. add-sqr-sqrt 57.1%

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

   13. sub-neg 57.1%

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

6. Applied egg-rr 57.1%

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

   1. associate-/r* 59.3%

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

8. Simplified 59.3%

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

9. Taylor expanded in t1 around inf 9.7%

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

10. Final simplification 9.7%

    \[\leadsto \frac{v}{t1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))