Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.1%

Time: 51.5s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% 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: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. associate-/l* 72.4%

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

   2. distribute-lft-neg-out 72.4%

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

   3. distribute-rgt-neg-in 72.4%

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

   4. associate-/r* 84.6%

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

   5. distribute-neg-frac2 84.6%

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

3. Simplified 84.6%

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

   1. associate-*r/ 99.1%

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

   2. neg-mul-1 99.1%

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

   3. associate-/r* 99.1%

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

6. Applied egg-rr 99.1%

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

   1. frac-2neg 99.1%

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

   2. metadata-eval 99.1%

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

   3. /-rgt-identity 99.1%

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

   4. *-commutative 99.1%

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

   5. distribute-rgt-neg-in 99.1%

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

8. Applied egg-rr 99.1%

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

9. Final simplification 99.1%

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

Alternative 2: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -2.3 \cdot 10^{+113}:\\ \;\;\;\;t\_2 \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+160}:\\ \;\;\;\;t1 \cdot \frac{t\_2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (/ v (+ t1 u))))
   (if (<= t1 -2.3e+113)
     (* t_2 (+ (/ u t1) -1.0))
     (if (<= t1 1.6e+160) (* t1 (/ t_2 t_1)) (/ v t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v / (t1 + u);
	double tmp;
	if (t1 <= -2.3e+113) {
		tmp = t_2 * ((u / t1) + -1.0);
	} else if (t1 <= 1.6e+160) {
		tmp = t1 * (t_2 / t_1);
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -u - t1
    t_2 = v / (t1 + u)
    if (t1 <= (-2.3d+113)) then
        tmp = t_2 * ((u / t1) + (-1.0d0))
    else if (t1 <= 1.6d+160) then
        tmp = t1 * (t_2 / t_1)
    else
        tmp = v / t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v / (t1 + u);
	double tmp;
	if (t1 <= -2.3e+113) {
		tmp = t_2 * ((u / t1) + -1.0);
	} else if (t1 <= 1.6e+160) {
		tmp = t1 * (t_2 / t_1);
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	t_2 = v / (t1 + u)
	tmp = 0
	if t1 <= -2.3e+113:
		tmp = t_2 * ((u / t1) + -1.0)
	elif t1 <= 1.6e+160:
		tmp = t1 * (t_2 / t_1)
	else:
		tmp = v / t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	t_2 = Float64(v / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -2.3e+113)
		tmp = Float64(t_2 * Float64(Float64(u / t1) + -1.0));
	elseif (t1 <= 1.6e+160)
		tmp = Float64(t1 * Float64(t_2 / t_1));
	else
		tmp = Float64(v / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	t_2 = v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -2.3e+113)
		tmp = t_2 * ((u / t1) + -1.0);
	elseif (t1 <= 1.6e+160)
		tmp = t1 * (t_2 / t_1);
	else
		tmp = v / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, Block[{t$95$2 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.3e+113], N[(t$95$2 * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.6e+160], N[(t1 * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(v / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.29999999999999997e113

   1. Initial program 51.3%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 95.5%

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

   if -2.29999999999999997e113 < t1 < 1.5999999999999999e160

   1. Initial program 80.5%

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

      1. associate-/l* 81.2%

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

      2. distribute-lft-neg-out 81.2%

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

      3. distribute-rgt-neg-in 81.2%

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

      4. associate-/r* 90.2%

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

      5. distribute-neg-frac2 90.2%

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

   3. Simplified 90.2%

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

   if 1.5999999999999999e160 < t1

   1. Initial program 52.7%

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

      1. associate-/l* 53.8%

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

      2. distribute-lft-neg-out 53.8%

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

      3. distribute-rgt-neg-in 53.8%

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

      4. associate-/r* 69.1%

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

      5. distribute-neg-frac2 69.1%

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

   3. Simplified 69.1%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/r* 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. frac-2neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. /-rgt-identity 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in t1 around inf 96.7%

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

       1. neg-mul-1 96.7%

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

   11. Simplified 96.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.3 \cdot 10^{+113}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+160}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 2.25 \cdot 10^{-24}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.1e-5)
   (* t1 (/ (/ v (+ t1 u)) (- u)))
   (if (<= u 2.25e-24) (- (/ v t1)) (/ (* t1 (/ v (- u))) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.1e-5) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else if (u <= 2.25e-24) {
		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 <= (-3.1d-5)) then
        tmp = t1 * ((v / (t1 + u)) / -u)
    else if (u <= 2.25d-24) 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 <= -3.1e-5) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else if (u <= 2.25e-24) {
		tmp = -(v / t1);
	} else {
		tmp = (t1 * (v / -u)) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.1e-5:
		tmp = t1 * ((v / (t1 + u)) / -u)
	elif u <= 2.25e-24:
		tmp = -(v / t1)
	else:
		tmp = (t1 * (v / -u)) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.1e-5)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(-u)));
	elseif (u <= 2.25e-24)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.1e-5)
		tmp = t1 * ((v / (t1 + u)) / -u);
	elseif (u <= 2.25e-24)
		tmp = -(v / t1);
	else
		tmp = (t1 * (v / -u)) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.1e-5], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.25e-24], (-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 < -3.10000000000000014e-5

   1. Initial program 79.2%

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

      1. associate-/l* 79.6%

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

      2. distribute-lft-neg-out 79.6%

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

      3. distribute-rgt-neg-in 79.6%

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

      4. associate-/r* 92.7%

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

      5. distribute-neg-frac2 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in t1 around 0 85.7%

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

      1. mul-1-neg 85.7%

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

   7. Simplified 85.7%

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

   if -3.10000000000000014e-5 < u < 2.2499999999999999e-24

   1. Initial program 67.3%

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

      1. associate-/l* 66.4%

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

      2. distribute-lft-neg-out 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. associate-/r* 77.7%

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

      5. distribute-neg-frac2 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   7. Simplified 86.5%

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

   if 2.2499999999999999e-24 < u

   1. Initial program 74.6%

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

      1. associate-/l* 75.5%

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

      2. distribute-lft-neg-out 75.5%

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

      3. distribute-rgt-neg-in 75.5%

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

      4. associate-/r* 88.2%

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

      5. distribute-neg-frac2 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in t1 around 0 75.5%

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

      1. mul-1-neg 75.5%

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

   7. Simplified 75.5%

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

   8. Taylor expanded in v around 0 68.0%

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

      1. mul-1-neg 68.0%

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

      2. associate-/r* 74.0%

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

      3. distribute-neg-frac 74.0%

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

      4. associate-/l* 76.9%

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

      5. distribute-rgt-neg-in 76.9%

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

      6. mul-1-neg 76.9%

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

      7. associate-*r/ 76.9%

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

      8. mul-1-neg 76.9%

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

   10. Simplified 76.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 2.25 \cdot 10^{-24}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5e-7)
   (* t1 (/ (/ v (+ t1 u)) (- u)))
   (if (<= u 2.3e-24) (- (/ v t1)) (/ (* t1 (/ v u)) (- t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e-7) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else if (u <= 2.3e-24) {
		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 <= (-5d-7)) then
        tmp = t1 * ((v / (t1 + u)) / -u)
    else if (u <= 2.3d-24) 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 <= -5e-7) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else if (u <= 2.3e-24) {
		tmp = -(v / t1);
	} else {
		tmp = (t1 * (v / u)) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5e-7:
		tmp = t1 * ((v / (t1 + u)) / -u)
	elif u <= 2.3e-24:
		tmp = -(v / t1)
	else:
		tmp = (t1 * (v / u)) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5e-7)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(-u)));
	elseif (u <= 2.3e-24)
		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 <= -5e-7)
		tmp = t1 * ((v / (t1 + u)) / -u);
	elseif (u <= 2.3e-24)
		tmp = -(v / t1);
	else
		tmp = (t1 * (v / u)) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5e-7], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.3e-24], (-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 < -4.99999999999999977e-7

   1. Initial program 79.2%

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

      1. associate-/l* 79.6%

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

      2. distribute-lft-neg-out 79.6%

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

      3. distribute-rgt-neg-in 79.6%

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

      4. associate-/r* 92.7%

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

      5. distribute-neg-frac2 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in t1 around 0 85.7%

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

      1. mul-1-neg 85.7%

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

   7. Simplified 85.7%

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

   if -4.99999999999999977e-7 < u < 2.3000000000000001e-24

   1. Initial program 67.3%

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

      1. associate-/l* 66.4%

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

      2. distribute-lft-neg-out 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. associate-/r* 77.7%

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

      5. distribute-neg-frac2 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   7. Simplified 86.5%

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

   if 2.3000000000000001e-24 < u

   1. Initial program 74.6%

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

      1. associate-/l* 75.5%

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

      2. distribute-lft-neg-out 75.5%

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

      3. distribute-rgt-neg-in 75.5%

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

      4. associate-/r* 88.2%

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

      5. distribute-neg-frac2 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in t1 around 0 75.5%

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

      1. mul-1-neg 75.5%

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

   7. Simplified 75.5%

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

      1. associate-*r/ 78.2%

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

      2. distribute-frac-neg2 78.2%

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

      3. add-sqr-sqrt 78.2%

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

      4. sqrt-unprod 70.1%

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

      5. sqr-neg 70.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 52.8%

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

      8. associate-*r/ 53.4%

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

      9. clear-num 53.5%

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

      10. un-div-inv 53.5%

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

      11. associate-/r/ 53.5%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 69.9%

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

      14. sqr-neg 69.9%

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

      15. sqrt-unprod 75.5%

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

      16. add-sqr-sqrt 75.5%

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

   9. Applied egg-rr 75.5%

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

       1. distribute-neg-frac2 75.5%

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

       2. *-commutative 75.5%

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

       3. distribute-rgt-neg-in 75.5%

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

   11. Simplified 75.5%

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

       1. associate-/r* 74.8%

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

       2. div-inv 74.8%

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

       3. distribute-neg-frac2 74.8%

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

       4. clear-num 74.8%

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

       5. times-frac 68.0%

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

       6. associate-/l/ 74.0%

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

       7. frac-2neg 74.0%

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

       8. div-inv 73.9%

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

   13. Applied egg-rr 74.3%

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

       1. associate-*r/ 74.4%

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

       2. *-rgt-identity 74.4%

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

       3. associate-*r/ 74.1%

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

       4. associate-*l/ 76.9%

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

       5. *-commutative 76.9%

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

   15. Simplified 76.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{-24}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.85e-5)
   (* t1 (/ (/ v (- u)) u))
   (if (<= u 1.55e-24) (- (/ v t1)) (/ (* t1 (/ v u)) (- t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.85e-5) {
		tmp = t1 * ((v / -u) / u);
	} else if (u <= 1.55e-24) {
		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 <= (-2.85d-5)) then
        tmp = t1 * ((v / -u) / u)
    else if (u <= 1.55d-24) 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 <= -2.85e-5) {
		tmp = t1 * ((v / -u) / u);
	} else if (u <= 1.55e-24) {
		tmp = -(v / t1);
	} else {
		tmp = (t1 * (v / u)) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.85e-5:
		tmp = t1 * ((v / -u) / u)
	elif u <= 1.55e-24:
		tmp = -(v / t1)
	else:
		tmp = (t1 * (v / u)) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.85e-5)
		tmp = Float64(t1 * Float64(Float64(v / Float64(-u)) / u));
	elseif (u <= 1.55e-24)
		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 <= -2.85e-5)
		tmp = t1 * ((v / -u) / u);
	elseif (u <= 1.55e-24)
		tmp = -(v / t1);
	else
		tmp = (t1 * (v / u)) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.85e-5], N[(t1 * N[(N[(v / (-u)), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.55e-24], (-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 < -2.8500000000000002e-5

   1. Initial program 79.2%

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

      1. associate-/l* 79.6%

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

      2. distribute-lft-neg-out 79.6%

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

      3. distribute-rgt-neg-in 79.6%

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

      4. associate-/r* 92.7%

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

      5. distribute-neg-frac2 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in t1 around 0 85.7%

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

      1. mul-1-neg 85.7%

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

   7. Simplified 85.7%

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

   8. Taylor expanded in t1 around 0 85.7%

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

   if -2.8500000000000002e-5 < u < 1.55e-24

   1. Initial program 67.3%

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

      1. associate-/l* 66.4%

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

      2. distribute-lft-neg-out 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. associate-/r* 77.7%

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

      5. distribute-neg-frac2 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   7. Simplified 86.5%

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

   if 1.55e-24 < u

   1. Initial program 74.6%

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

      1. associate-/l* 75.5%

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

      2. distribute-lft-neg-out 75.5%

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

      3. distribute-rgt-neg-in 75.5%

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

      4. associate-/r* 88.2%

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

      5. distribute-neg-frac2 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in t1 around 0 75.5%

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

      1. mul-1-neg 75.5%

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

   7. Simplified 75.5%

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

      1. associate-*r/ 78.2%

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

      2. distribute-frac-neg2 78.2%

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

      3. add-sqr-sqrt 78.2%

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

      4. sqrt-unprod 70.1%

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

      5. sqr-neg 70.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 52.8%

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

      8. associate-*r/ 53.4%

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

      9. clear-num 53.5%

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

      10. un-div-inv 53.5%

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

      11. associate-/r/ 53.5%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 69.9%

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

      14. sqr-neg 69.9%

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

      15. sqrt-unprod 75.5%

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

      16. add-sqr-sqrt 75.5%

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

   9. Applied egg-rr 75.5%

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

       1. distribute-neg-frac2 75.5%

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

       2. *-commutative 75.5%

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

       3. distribute-rgt-neg-in 75.5%

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

   11. Simplified 75.5%

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

       1. associate-/r* 74.8%

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

       2. div-inv 74.8%

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

       3. distribute-neg-frac2 74.8%

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

       4. clear-num 74.8%

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

       5. times-frac 68.0%

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

       6. associate-/l/ 74.0%

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

       7. frac-2neg 74.0%

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

       8. div-inv 73.9%

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

   13. Applied egg-rr 74.3%

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

       1. associate-*r/ 74.4%

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

       2. *-rgt-identity 74.4%

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

       3. associate-*r/ 74.1%

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

       4. associate-*l/ 76.9%

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

       5. *-commutative 76.9%

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

   15. Simplified 76.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{-24}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -0.00047 \lor \neg \left(u \leq 10^{-24}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -0.00047) (not (<= u 1e-24)))
   (* t1 (/ (/ v (- u)) u))
   (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -0.00047) || !(u <= 1e-24)) {
		tmp = t1 * ((v / -u) / u);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-0.00047d0)) .or. (.not. (u <= 1d-24))) then
        tmp = t1 * ((v / -u) / u)
    else
        tmp = -(v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -0.00047) || !(u <= 1e-24)) {
		tmp = t1 * ((v / -u) / u);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -0.00047) or not (u <= 1e-24):
		tmp = t1 * ((v / -u) / u)
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -0.00047) || !(u <= 1e-24))
		tmp = Float64(t1 * Float64(Float64(v / Float64(-u)) / u));
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -0.00047) || ~((u <= 1e-24)))
		tmp = t1 * ((v / -u) / u);
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -0.00047], N[Not[LessEqual[u, 1e-24]], $MachinePrecision]], N[(t1 * N[(N[(v / (-u)), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -4.69999999999999986e-4 or 9.99999999999999924e-25 < u

   1. Initial program 76.7%

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

      1. associate-/l* 77.4%

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

      2. distribute-lft-neg-out 77.4%

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

      3. distribute-rgt-neg-in 77.4%

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

      4. associate-/r* 90.3%

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

      5. distribute-neg-frac2 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in t1 around 0 80.2%

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

      1. mul-1-neg 80.2%

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

   7. Simplified 80.2%

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

   8. Taylor expanded in t1 around 0 79.6%

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

   if -4.69999999999999986e-4 < u < 9.99999999999999924e-25

   1. Initial program 67.3%

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

      1. associate-/l* 66.4%

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

      2. distribute-lft-neg-out 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. associate-/r* 77.7%

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

      5. distribute-neg-frac2 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   7. Simplified 86.5%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.00047 \lor \neg \left(u \leq 10^{-24}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.05 \cdot 10^{+53} \lor \neg \left(u \leq 2.1 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.05e+53) (not (<= u 2.1e+103)))
   (/ t1 (* u (/ u v)))
   (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.05e+53) || !(u <= 2.1e+103)) {
		tmp = 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 <= (-1.05d+53)) .or. (.not. (u <= 2.1d+103))) then
        tmp = 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 <= -1.05e+53) || !(u <= 2.1e+103)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.05e+53) or not (u <= 2.1e+103):
		tmp = t1 / (u * (u / v))
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.05e+53) || !(u <= 2.1e+103))
		tmp = Float64(t1 / Float64(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 <= -1.05e+53) || ~((u <= 2.1e+103)))
		tmp = t1 / (u * (u / v));
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.05e+53], N[Not[LessEqual[u, 2.1e+103]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -1.0500000000000001e53 or 2.1000000000000002e103 < u

   1. Initial program 79.4%

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

      1. associate-/l* 78.9%

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

      2. distribute-lft-neg-out 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

      4. associate-/r* 90.7%

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

      5. distribute-neg-frac2 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t1 around 0 86.6%

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

      1. mul-1-neg 86.6%

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

   7. Simplified 86.6%

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

      1. associate-*r/ 89.5%

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

      2. distribute-frac-neg2 89.5%

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

      3. add-sqr-sqrt 43.0%

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

      4. sqrt-unprod 75.4%

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

      5. sqr-neg 75.4%

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

      6. sqrt-unprod 38.1%

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

      7. add-sqr-sqrt 71.0%

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

      8. associate-*r/ 71.7%

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

      9. clear-num 71.7%

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

      10. un-div-inv 71.7%

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

      11. associate-/r/ 71.7%

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

      12. add-sqr-sqrt 38.4%

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

      13. sqrt-unprod 75.3%

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

      14. sqr-neg 75.3%

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

      15. sqrt-unprod 40.8%

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

      16. add-sqr-sqrt 85.8%

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

   9. Applied egg-rr 85.8%

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

       1. distribute-neg-frac2 85.8%

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

       2. *-commutative 85.8%

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

       3. distribute-rgt-neg-in 85.8%

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

   11. Simplified 85.8%

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

   12. Taylor expanded in t1 around 0 85.9%

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

       1. distribute-rgt-neg-out 85.9%

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

       2. distribute-lft-neg-in 85.9%

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

       3. add-sqr-sqrt 45.0%

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

       4. sqrt-unprod 73.6%

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

       5. sqr-neg 73.6%

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

       6. sqrt-unprod 33.1%

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

       7. add-sqr-sqrt 71.4%

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

       8. clear-num 71.4%

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

       9. un-div-inv 71.4%

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

   14. Applied egg-rr 71.4%

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

       1. associate-/r/ 71.4%

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

   16. Simplified 71.4%

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

   if -1.0500000000000001e53 < u < 2.1000000000000002e103

   1. Initial program 67.5%

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

      1. associate-/l* 67.8%

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

      2. distribute-lft-neg-out 67.8%

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

      3. distribute-rgt-neg-in 67.8%

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

      4. associate-/r* 80.3%

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

      5. distribute-neg-frac2 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in t1 around inf 77.3%

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

      1. associate-*r/ 77.3%

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

      2. neg-mul-1 77.3%

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

   7. Simplified 77.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.05 \cdot 10^{+53} \lor \neg \left(u \leq 2.1 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.3 \cdot 10^{+152} \lor \neg \left(u \leq 4.7 \cdot 10^{+104}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.3e+152) (not (<= u 4.7e+104)))
   (* v (/ (/ t1 u) u))
   (/ v (- (- u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.3e+152) || !(u <= 4.7e+104)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.3d+152)) .or. (.not. (u <= 4.7d+104))) then
        tmp = v * ((t1 / u) / u)
    else
        tmp = v / (-u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.3e+152) || !(u <= 4.7e+104)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.3e+152) or not (u <= 4.7e+104):
		tmp = v * ((t1 / u) / u)
	else:
		tmp = v / (-u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.3e+152) || !(u <= 4.7e+104))
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	else
		tmp = Float64(v / Float64(Float64(-u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.3e+152) || ~((u <= 4.7e+104)))
		tmp = v * ((t1 / u) / u);
	else
		tmp = v / (-u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.3e+152], N[Not[LessEqual[u, 4.7e+104]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.29999999999999976e152 or 4.70000000000000017e104 < u

   1. Initial program 78.5%

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

      1. associate-/l* 77.8%

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

      2. distribute-lft-neg-out 77.8%

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

      3. distribute-rgt-neg-in 77.8%

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

      4. associate-/r* 92.4%

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

      5. distribute-neg-frac2 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in t1 around 0 88.4%

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

      1. mul-1-neg 88.4%

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

   7. Simplified 88.4%

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

      1. associate-*r/ 91.3%

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

      2. distribute-frac-neg2 91.3%

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

      3. add-sqr-sqrt 53.6%

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

      4. sqrt-unprod 77.2%

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

      5. sqr-neg 77.2%

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

      6. sqrt-unprod 30.6%

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

      7. add-sqr-sqrt 71.6%

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

      8. associate-*r/ 72.2%

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

      9. clear-num 72.2%

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

      10. un-div-inv 72.2%

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

      11. associate-/r/ 72.2%

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

      12. add-sqr-sqrt 30.6%

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

      13. sqrt-unprod 76.7%

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

      14. sqr-neg 76.7%

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

      15. sqrt-unprod 50.9%

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

      16. add-sqr-sqrt 87.5%

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

   9. Applied egg-rr 87.5%

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

       1. distribute-neg-frac2 87.5%

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

       2. *-commutative 87.5%

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

       3. distribute-rgt-neg-in 87.5%

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

   11. Simplified 87.5%

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

   12. Taylor expanded in t1 around 0 87.6%

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

       1. associate-/r* 88.8%

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

       2. distribute-neg-frac 88.8%

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

       3. associate-/r/ 77.8%

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

       4. add-sqr-sqrt 33.0%

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

       5. sqrt-unprod 71.8%

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

       6. sqr-neg 71.8%

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

       7. sqrt-unprod 40.9%

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

       8. add-sqr-sqrt 71.6%

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

   14. Applied egg-rr 71.6%

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

   if -5.29999999999999976e152 < u < 4.70000000000000017e104

   1. Initial program 69.4%

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

      1. associate-/l* 69.7%

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

      2. distribute-lft-neg-out 69.7%

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

      3. distribute-rgt-neg-in 69.7%

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

      4. associate-/r* 80.7%

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

      5. distribute-neg-frac2 80.7%

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

   3. Simplified 80.7%

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

      1. associate-*r/ 98.7%

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

      2. neg-mul-1 98.7%

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

      3. associate-/r* 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. frac-2neg 98.7%

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

      2. metadata-eval 98.7%

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

      3. /-rgt-identity 98.7%

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

      4. *-commutative 98.7%

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

      5. distribute-rgt-neg-in 98.7%

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

   8. Applied egg-rr 98.7%

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

   9. Taylor expanded in t1 around inf 75.5%

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

       1. neg-mul-1 75.5%

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

   11. Simplified 75.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.3 \cdot 10^{+152} \lor \neg \left(u \leq 4.7 \cdot 10^{+104}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+152} \lor \neg \left(u \leq 1.55 \cdot 10^{+156}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.8e+152) (not (<= u 1.55e+156))) (/ v u) (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.8e+152) || !(u <= 1.55e+156)) {
		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 <= (-3.8d+152)) .or. (.not. (u <= 1.55d+156))) 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 <= -3.8e+152) || !(u <= 1.55e+156)) {
		tmp = v / u;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.8e+152) or not (u <= 1.55e+156):
		tmp = v / u
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.8e+152) || !(u <= 1.55e+156))
		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 <= -3.8e+152) || ~((u <= 1.55e+156)))
		tmp = v / u;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.8e+152], N[Not[LessEqual[u, 1.55e+156]], $MachinePrecision]], N[(v / u), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -3.8e152 or 1.5500000000000001e156 < u

   1. Initial program 78.1%

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

      1. associate-/l* 78.4%

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

      2. distribute-lft-neg-out 78.4%

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

      3. distribute-rgt-neg-in 78.4%

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

      4. associate-/r* 95.5%

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

      5. distribute-neg-frac2 95.5%

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

   3. Simplified 95.5%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/r* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 48.8%

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

      1. div-inv 48.8%

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

      2. frac-2neg 48.8%

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

      3. metadata-eval 48.8%

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

      4. /-rgt-identity 48.8%

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

      5. add-sqr-sqrt 26.6%

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

      6. sqrt-unprod 46.9%

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

      7. sqr-neg 46.9%

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

      8. sqrt-unprod 20.7%

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

      9. add-sqr-sqrt 46.1%

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

      10. frac-2neg 46.1%

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

      11. metadata-eval 46.1%

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

      12. distribute-neg-in 46.1%

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

      13. add-sqr-sqrt 23.8%

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

      14. sqrt-unprod 46.0%

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

      15. sqr-neg 46.0%

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

      16. sqrt-unprod 22.3%

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

      17. add-sqr-sqrt 49.0%

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

      18. sub-neg 49.0%

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

   9. Applied egg-rr 49.0%

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

       1. associate-*r/ 49.0%

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

       2. *-commutative 49.0%

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

       3. neg-mul-1 49.0%

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

   11. Simplified 49.0%

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

   12. Taylor expanded in t1 around 0 46.2%

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

   if -3.8e152 < u < 1.5500000000000001e156

   1. Initial program 70.4%

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

      1. associate-/l* 70.3%

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

      2. distribute-lft-neg-out 70.3%

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

      3. distribute-rgt-neg-in 70.3%

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

      4. associate-/r* 80.7%

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

      5. distribute-neg-frac2 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in t1 around inf 69.7%

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

      1. associate-*r/ 69.7%

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

      2. neg-mul-1 69.7%

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

   7. Simplified 69.7%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+152} \lor \neg \left(u \leq 1.55 \cdot 10^{+156}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{+156}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.4e+50)
   (/ 1.0 (/ u v))
   (if (<= u 1.05e+156) (- (/ v t1)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+50) {
		tmp = 1.0 / (u / v);
	} else if (u <= 1.05e+156) {
		tmp = -(v / t1);
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.4d+50)) then
        tmp = 1.0d0 / (u / v)
    else if (u <= 1.05d+156) then
        tmp = -(v / t1)
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+50) {
		tmp = 1.0 / (u / v);
	} else if (u <= 1.05e+156) {
		tmp = -(v / t1);
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.4e+50:
		tmp = 1.0 / (u / v)
	elif u <= 1.05e+156:
		tmp = -(v / t1)
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.4e+50)
		tmp = Float64(1.0 / Float64(u / v));
	elseif (u <= 1.05e+156)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.4e+50)
		tmp = 1.0 / (u / v);
	elseif (u <= 1.05e+156)
		tmp = -(v / t1);
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.4e+50], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.05e+156], (-N[(v / t1), $MachinePrecision]), N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.3999999999999999e50

   1. Initial program 80.9%

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

      1. associate-/l* 81.2%

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

      2. distribute-lft-neg-out 81.2%

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

      3. distribute-rgt-neg-in 81.2%

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

      4. associate-/r* 91.7%

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

      5. distribute-neg-frac2 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in t1 around 0 90.0%

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

      1. mul-1-neg 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in t1 around inf 46.3%

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

      1. associate-*r/ 46.3%

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

      2. mul-1-neg 46.3%

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

   10. Simplified 46.3%

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

       1. clear-num 48.5%

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

       2. distribute-neg-frac2 48.5%

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

       3. inv-pow 48.5%

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

       4. add-sqr-sqrt 19.3%

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

       5. sqrt-unprod 69.1%

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

       6. sqr-neg 69.1%

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

       7. sqrt-unprod 29.0%

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

       8. add-sqr-sqrt 48.4%

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

   12. Applied egg-rr 48.4%

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

       1. unpow-1 48.4%

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

   14. Simplified 48.4%

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

   if -1.3999999999999999e50 < u < 1.04999999999999991e156

   1. Initial program 68.7%

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

      1. associate-/l* 68.4%

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

      2. distribute-lft-neg-out 68.4%

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

      3. distribute-rgt-neg-in 68.4%

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

      4. associate-/r* 80.3%

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

      5. distribute-neg-frac2 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in t1 around inf 73.1%

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

      1. associate-*r/ 73.1%

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

      2. neg-mul-1 73.1%

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

   7. Simplified 73.1%

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

   if 1.04999999999999991e156 < u

   1. Initial program 77.3%

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

      1. associate-/l* 77.6%

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

      2. distribute-lft-neg-out 77.6%

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

      3. distribute-rgt-neg-in 77.6%

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

      4. associate-/r* 94.3%

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

      5. distribute-neg-frac2 94.3%

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

   3. Simplified 94.3%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/r* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 43.7%

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

      1. div-inv 43.7%

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

      2. frac-2neg 43.7%

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

      3. metadata-eval 43.7%

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

      4. /-rgt-identity 43.7%

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

      5. add-sqr-sqrt 21.7%

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

      6. sqrt-unprod 40.2%

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

      7. sqr-neg 40.2%

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

      8. sqrt-unprod 19.0%

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

      9. add-sqr-sqrt 41.1%

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

      10. frac-2neg 41.1%

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

      11. metadata-eval 41.1%

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

      12. distribute-neg-in 41.1%

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

      13. add-sqr-sqrt 25.0%

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

      14. sqrt-unprod 40.9%

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

      15. sqr-neg 40.9%

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

      16. sqrt-unprod 16.1%

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

      17. add-sqr-sqrt 44.0%

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

      18. sub-neg 44.0%

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

   9. Applied egg-rr 44.0%

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

       1. associate-*r/ 44.0%

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

       2. *-commutative 44.0%

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

       3. neg-mul-1 44.0%

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

   11. Simplified 44.0%

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

   12. Taylor expanded in t1 around 0 41.2%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{+156}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 23.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{+99} \lor \neg \left(t1 \leq 7.5 \cdot 10^{+72}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.9e+99) (not (<= t1 7.5e+72))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.9e+99) || !(t1 <= 7.5e+72)) {
		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 <= (-2.9d+99)) .or. (.not. (t1 <= 7.5d+72))) 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 <= -2.9e+99) || !(t1 <= 7.5e+72)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.9e+99) or not (t1 <= 7.5e+72):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.9e+99) || !(t1 <= 7.5e+72))
		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 <= -2.9e+99) || ~((t1 <= 7.5e+72)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.9e+99], N[Not[LessEqual[t1, 7.5e+72]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.9000000000000002e99 or 7.50000000000000027e72 < t1

   1. Initial program 54.5%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 93.4%

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

   6. Taylor expanded in u around inf 40.0%

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

   if -2.9000000000000002e99 < t1 < 7.50000000000000027e72

   1. Initial program 82.0%

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

      1. associate-/l* 81.5%

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

      2. distribute-lft-neg-out 81.5%

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

      3. distribute-rgt-neg-in 81.5%

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

      4. associate-/r* 90.2%

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

      5. distribute-neg-frac2 90.2%

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

   3. Simplified 90.2%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

      3. associate-/r* 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in t1 around inf 51.0%

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

      1. div-inv 50.8%

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

      2. frac-2neg 50.8%

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

      3. metadata-eval 50.8%

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

      4. /-rgt-identity 50.8%

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

      5. add-sqr-sqrt 27.2%

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

      6. sqrt-unprod 32.1%

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

      7. sqr-neg 32.1%

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

      8. sqrt-unprod 8.8%

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

      9. add-sqr-sqrt 19.7%

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

      10. frac-2neg 19.7%

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

      11. metadata-eval 19.7%

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

      12. distribute-neg-in 19.7%

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

      13. add-sqr-sqrt 10.3%

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

      14. sqrt-unprod 33.4%

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

      15. sqr-neg 33.4%

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

      16. sqrt-unprod 25.3%

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

      17. add-sqr-sqrt 50.8%

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

      18. sub-neg 50.8%

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

   9. Applied egg-rr 50.8%

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

       1. associate-*r/ 51.0%

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

       2. *-commutative 51.0%

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

       3. neg-mul-1 51.0%

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

   11. Simplified 51.0%

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

   12. Taylor expanded in t1 around 0 21.8%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{+99} \lor \neg \left(t1 \leq 7.5 \cdot 10^{+72}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. associate-/l* 72.4%

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

   2. distribute-lft-neg-out 72.4%

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

   3. distribute-rgt-neg-in 72.4%

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

   4. associate-/r* 84.6%

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

   5. distribute-neg-frac2 84.6%

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

3. Simplified 84.6%

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

   1. distribute-frac-neg2 84.6%

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

   2. distribute-rgt-neg-out 84.6%

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

   3. associate-/r* 72.4%

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

   4. distribute-lft-neg-out 72.4%

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

   5. associate-/l* 72.5%

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

   6. times-frac 97.9%

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

   7. frac-2neg 97.9%

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

   8. associate-*r/ 98.6%

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

   9. add-sqr-sqrt 49.3%

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

   10. sqrt-unprod 46.6%

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

   11. sqr-neg 46.6%

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

   12. sqrt-unprod 23.5%

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

   13. add-sqr-sqrt 41.0%

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

   14. add-sqr-sqrt 19.3%

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

   15. sqrt-unprod 59.1%

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

   16. sqr-neg 59.1%

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

   17. sqrt-prod 50.1%

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

   18. add-sqr-sqrt 98.6%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 13: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. times-frac 97.9%

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

   2. distribute-frac-neg 97.9%

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

   3. distribute-neg-frac2 97.9%

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

   4. +-commutative 97.9%

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

   5. distribute-neg-in 97.9%

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

   6. unsub-neg 97.9%

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

3. Simplified 97.9%

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

5. Final simplification 97.9%

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

Alternative 14: 61.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. associate-/l* 72.4%

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

   2. distribute-lft-neg-out 72.4%

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

   3. distribute-rgt-neg-in 72.4%

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

   4. associate-/r* 84.6%

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

   5. distribute-neg-frac2 84.6%

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

3. Simplified 84.6%

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

   1. associate-*r/ 99.1%

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

   2. neg-mul-1 99.1%

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

   3. associate-/r* 99.1%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in t1 around inf 65.7%

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

   1. div-inv 65.5%

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

   2. frac-2neg 65.5%

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

   3. metadata-eval 65.5%

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

   4. /-rgt-identity 65.5%

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

   5. add-sqr-sqrt 35.7%

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

   6. sqrt-unprod 41.4%

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

   7. sqr-neg 41.4%

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

   8. sqrt-unprod 12.7%

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

   9. add-sqr-sqrt 27.2%

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

   10. frac-2neg 27.2%

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

   11. metadata-eval 27.2%

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

   12. distribute-neg-in 27.2%

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

   13. add-sqr-sqrt 12.8%

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

   14. sqrt-unprod 38.9%

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

   15. sqr-neg 38.9%

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

   16. sqrt-unprod 32.1%

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

   17. add-sqr-sqrt 65.5%

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

   18. sub-neg 65.5%

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

9. Applied egg-rr 65.5%

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

    1. associate-*r/ 65.7%

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

    2. *-commutative 65.7%

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

    3. neg-mul-1 65.7%

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

11. Simplified 65.7%

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

    1. frac-2neg 65.7%

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

    2. div-inv 65.5%

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

    3. remove-double-neg 65.5%

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

    4. sub-neg 65.5%

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

    5. distribute-neg-in 65.5%

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

    6. add-sqr-sqrt 33.9%

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

    7. sqrt-unprod 72.9%

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

    8. sqr-neg 72.9%

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

    9. sqrt-unprod 31.1%

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

    10. add-sqr-sqrt 65.5%

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

    11. distribute-neg-in 65.5%

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

    12. +-commutative 65.5%

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

    13. distribute-neg-in 65.5%

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

    14. add-sqr-sqrt 34.4%

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

    15. sqrt-unprod 73.7%

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

    16. sqr-neg 73.7%

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

    17. sqrt-unprod 31.6%

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

    18. add-sqr-sqrt 65.5%

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

13. Applied egg-rr 65.5%

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

    1. associate-*r/ 65.7%

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

    2. *-rgt-identity 65.7%

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

    3. sub-neg 65.7%

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

15. Simplified 65.7%

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

Alternative 15: 14.0% 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 72.5%

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

   1. times-frac 97.9%

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

   2. distribute-frac-neg 97.9%

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

   3. distribute-neg-frac2 97.9%

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

   4. +-commutative 97.9%

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

   5. distribute-neg-in 97.9%

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

   6. unsub-neg 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in t1 around inf 58.8%

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

6. Taylor expanded in u around inf 15.5%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))