Rosa's DopplerBench

Percentage Accurate: 71.9% → 97.9%

Time: 9.6s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. associate-/r* 83.9%

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

   2. associate-/l* 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 89.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{u}{\frac{t1}{v}} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -3.6 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{-162}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -1.4e+154)
     (/ (- (/ u (/ t1 v)) v) (+ t1 u))
     (if (<= t1 -3.6e-147)
       t_1
       (if (<= t1 2.4e-162)
         (* (/ (- v) u) (/ t1 u))
         (if (<= t1 1.5e+126) t_1 (/ (- v) (+ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.4e+154) {
		tmp = ((u / (t1 / v)) - v) / (t1 + u);
	} else if (t1 <= -3.6e-147) {
		tmp = t_1;
	} else if (t1 <= 2.4e-162) {
		tmp = (-v / u) * (t1 / u);
	} else if (t1 <= 1.5e+126) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-1.4d+154)) then
        tmp = ((u / (t1 / v)) - v) / (t1 + u)
    else if (t1 <= (-3.6d-147)) then
        tmp = t_1
    else if (t1 <= 2.4d-162) then
        tmp = (-v / u) * (t1 / u)
    else if (t1 <= 1.5d+126) then
        tmp = t_1
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.4e+154) {
		tmp = ((u / (t1 / v)) - v) / (t1 + u);
	} else if (t1 <= -3.6e-147) {
		tmp = t_1;
	} else if (t1 <= 2.4e-162) {
		tmp = (-v / u) * (t1 / u);
	} else if (t1 <= 1.5e+126) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -1.4e+154:
		tmp = ((u / (t1 / v)) - v) / (t1 + u)
	elif t1 <= -3.6e-147:
		tmp = t_1
	elif t1 <= 2.4e-162:
		tmp = (-v / u) * (t1 / u)
	elif t1 <= 1.5e+126:
		tmp = t_1
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -1.4e+154)
		tmp = Float64(Float64(Float64(u / Float64(t1 / v)) - v) / Float64(t1 + u));
	elseif (t1 <= -3.6e-147)
		tmp = t_1;
	elseif (t1 <= 2.4e-162)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	elseif (t1 <= 1.5e+126)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -1.4e+154)
		tmp = ((u / (t1 / v)) - v) / (t1 + u);
	elseif (t1 <= -3.6e-147)
		tmp = t_1;
	elseif (t1 <= 2.4e-162)
		tmp = (-v / u) * (t1 / u);
	elseif (t1 <= 1.5e+126)
		tmp = t_1;
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.4e+154], N[(N[(N[(u / N[(t1 / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -3.6e-147], t$95$1, If[LessEqual[t1, 2.4e-162], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.5e+126], t$95$1, N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.4e154

   1. Initial program 39.6%

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

      1. associate-/r* 74.8%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 78.3%

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

      1. neg-mul-1 78.3%

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

      2. +-commutative 78.3%

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

      3. unsub-neg 78.3%

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

      4. associate-/l* 89.3%

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

   6. Simplified 89.3%

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

   if -1.4e154 < t1 < -3.60000000000000012e-147 or 2.4000000000000002e-162 < t1 < 1.5000000000000001e126

   1. Initial program 87.1%

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

      1. associate-*l/ 95.0%

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

      2. *-commutative 95.0%

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

   3. Simplified 95.0%

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

   if -3.60000000000000012e-147 < t1 < 2.4000000000000002e-162

   1. Initial program 73.4%

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

      1. associate-*l/ 71.9%

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

      2. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in t1 around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

      3. unpow2 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in v around 0 73.4%

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

      1. unpow2 73.4%

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

      2. *-commutative 73.4%

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

      3. times-frac 85.0%

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

      4. associate-*r* 85.0%

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

      5. associate-*r/ 85.0%

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

      6. neg-mul-1 85.0%

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

   9. Simplified 85.0%

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

   if 1.5000000000000001e126 < t1

   1. Initial program 48.5%

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

      1. associate-/r* 68.5%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around inf 91.7%

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

      1. neg-mul-1 91.7%

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

   6. Simplified 91.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{u}{\frac{t1}{v}} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -3.6 \cdot 10^{-147}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{-162}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{+126}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 3: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.05 \cdot 10^{-56}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 2.25 \cdot 10^{-60}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.05e-56)
   (/ (* t1 (/ v u)) (- t1 u))
   (if (<= u 2.25e-60) (/ (- v) t1) (* (/ t1 u) (/ (- v) (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.05e-56) {
		tmp = (t1 * (v / u)) / (t1 - u);
	} else if (u <= 2.25e-60) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / u) * (-v / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.05d-56)) then
        tmp = (t1 * (v / u)) / (t1 - u)
    else if (u <= 2.25d-60) then
        tmp = -v / t1
    else
        tmp = (t1 / u) * (-v / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.05e-56) {
		tmp = (t1 * (v / u)) / (t1 - u);
	} else if (u <= 2.25e-60) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / u) * (-v / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.05e-56:
		tmp = (t1 * (v / u)) / (t1 - u)
	elif u <= 2.25e-60:
		tmp = -v / t1
	else:
		tmp = (t1 / u) * (-v / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.05e-56)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u));
	elseif (u <= 2.25e-60)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.05e-56)
		tmp = (t1 * (v / u)) / (t1 - u);
	elseif (u <= 2.25e-60)
		tmp = -v / t1;
	else
		tmp = (t1 / u) * (-v / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.05e-56], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.25e-60], N[((-v) / t1), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.05000000000000003e-56

   1. Initial program 75.6%

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

      1. associate-/r* 88.3%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 74.2%

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

      1. mul-1-neg 74.2%

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

      2. associate-/l* 77.7%

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

      3. distribute-neg-frac 77.7%

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

   6. Simplified 77.7%

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

      1. frac-2neg 77.7%

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

      2. div-inv 77.6%

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

      3. distribute-neg-frac 77.6%

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

      4. remove-double-neg 77.6%

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

      5. div-inv 77.6%

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

      6. clear-num 77.1%

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

      7. distribute-neg-in 77.1%

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

      8. add-sqr-sqrt 41.1%

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

      9. sqrt-unprod 74.4%

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

      10. sqr-neg 74.4%

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

      11. sqrt-unprod 36.4%

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

      12. add-sqr-sqrt 77.1%

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

      13. sub-neg 77.1%

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

   8. Applied egg-rr 77.1%

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

      1. associate-*r/ 77.1%

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

      2. *-rgt-identity 77.1%

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

   10. Simplified 77.1%

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

   if -1.05000000000000003e-56 < u < 2.25e-60

   1. Initial program 65.5%

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

      1. associate-*l/ 75.8%

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

      2. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in t1 around inf 87.5%

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

      1. associate-*r/ 87.5%

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

      2. neg-mul-1 87.5%

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

   6. Simplified 87.5%

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

   if 2.25e-60 < u

   1. Initial program 82.5%

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

      1. associate-/r* 87.8%

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

      2. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in t1 around 0 74.3%

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

      1. mul-1-neg 74.3%

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

      2. associate-/l* 75.9%

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

      3. distribute-neg-frac 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in v around 0 73.1%

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

      1. mul-1-neg 73.1%

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

      2. times-frac 83.3%

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

      3. distribute-rgt-neg-in 83.3%

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

      4. distribute-neg-frac 83.3%

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

   9. Simplified 83.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.05 \cdot 10^{-56}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 2.25 \cdot 10^{-60}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{t1 + u}\\ \end{array} \]

Alternative 4: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;u \leq 4.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.7e-56)
   (/ (/ (- t1) (/ u v)) (+ t1 u))
   (if (<= u 4.6e-63) (/ (- v) t1) (* (/ t1 u) (/ (- v) (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.7e-56) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (u <= 4.6e-63) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / u) * (-v / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.7d-56)) then
        tmp = (-t1 / (u / v)) / (t1 + u)
    else if (u <= 4.6d-63) then
        tmp = -v / t1
    else
        tmp = (t1 / u) * (-v / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.7e-56) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (u <= 4.6e-63) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / u) * (-v / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.7e-56:
		tmp = (-t1 / (u / v)) / (t1 + u)
	elif u <= 4.6e-63:
		tmp = -v / t1
	else:
		tmp = (t1 / u) * (-v / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.7e-56)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / Float64(t1 + u));
	elseif (u <= 4.6e-63)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.7e-56)
		tmp = (-t1 / (u / v)) / (t1 + u);
	elseif (u <= 4.6e-63)
		tmp = -v / t1;
	else
		tmp = (t1 / u) * (-v / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.7e-56], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.6e-63], N[((-v) / t1), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.69999999999999991e-56

   1. Initial program 75.6%

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

      1. associate-/r* 88.3%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 74.2%

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

      1. mul-1-neg 74.2%

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

      2. associate-/l* 77.7%

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

      3. distribute-neg-frac 77.7%

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

   6. Simplified 77.7%

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

   if -1.69999999999999991e-56 < u < 4.6e-63

   1. Initial program 65.5%

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

      1. associate-*l/ 75.8%

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

      2. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in t1 around inf 87.5%

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

      1. associate-*r/ 87.5%

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

      2. neg-mul-1 87.5%

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

   6. Simplified 87.5%

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

   if 4.6e-63 < u

   1. Initial program 82.5%

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

      1. associate-/r* 87.8%

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

      2. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in t1 around 0 74.3%

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

      1. mul-1-neg 74.3%

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

      2. associate-/l* 75.9%

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

      3. distribute-neg-frac 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in v around 0 73.1%

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

      1. mul-1-neg 73.1%

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

      2. times-frac 83.3%

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

      3. distribute-rgt-neg-in 83.3%

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

      4. distribute-neg-frac 83.3%

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

   9. Simplified 83.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;u \leq 4.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{t1 + u}\\ \end{array} \]

Alternative 5: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-16}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.05e+17)
   (/ (- v) (+ t1 u))
   (if (<= t1 4.6e-16) (* t1 (/ (/ (- v) u) u)) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.05e+17) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 4.6e-16) {
		tmp = t1 * ((-v / u) / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.05d+17)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 4.6d-16) then
        tmp = t1 * ((-v / u) / u)
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.05e+17) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 4.6e-16) {
		tmp = t1 * ((-v / u) / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.05e+17:
		tmp = -v / (t1 + u)
	elif t1 <= 4.6e-16:
		tmp = t1 * ((-v / u) / u)
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.05e+17)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 4.6e-16)
		tmp = Float64(t1 * Float64(Float64(Float64(-v) / u) / u));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.05e+17)
		tmp = -v / (t1 + u);
	elseif (t1 <= 4.6e-16)
		tmp = t1 * ((-v / u) / u);
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.05e+17], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.6e-16], N[(t1 * N[(N[((-v) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.05e17

   1. Initial program 51.0%

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

      1. associate-/r* 73.6%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 83.3%

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

      1. neg-mul-1 83.3%

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

   6. Simplified 83.3%

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

   if -1.05e17 < t1 < 4.5999999999999998e-16

   1. Initial program 83.6%

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

      1. associate-*l/ 85.8%

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

      2. *-commutative 85.8%

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

   3. Simplified 85.8%

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

      1. neg-mul-1 85.8%

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

      2. times-frac 94.6%

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

   5. Applied egg-rr 94.6%

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

      1. associate-*l/ 94.7%

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

      2. mul-1-neg 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in t1 around 0 73.5%

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

      1. associate-*r/ 75.4%

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

      2. unpow2 75.4%

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

      3. associate-/l/ 77.5%

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

      4. neg-mul-1 77.5%

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

      5. distribute-rgt-neg-in 77.5%

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

      6. distribute-neg-frac 77.5%

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

      7. distribute-neg-frac 77.5%

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

   10. Simplified 77.5%

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

   if 4.5999999999999998e-16 < t1

   1. Initial program 70.8%

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

      1. associate-*l/ 77.0%

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

      2. *-commutative 77.0%

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

   3. Simplified 77.0%

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

      1. neg-mul-1 77.0%

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

      2. times-frac 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. associate-*l/ 98.2%

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

      2. mul-1-neg 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in t1 around inf 83.7%

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

      1. *-commutative 83.7%

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

      2. associate-*l/ 84.0%

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

      3. metadata-eval 84.0%

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

      4. neg-mul-1 84.0%

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

      5. frac-2neg 84.0%

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

      6. frac-2neg 84.0%

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

      7. remove-double-neg 84.0%

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

      8. add-sqr-sqrt 39.1%

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

      9. sqrt-unprod 42.1%

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

      10. sqr-neg 42.1%

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

      11. sqrt-unprod 11.1%

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

      12. add-sqr-sqrt 25.4%

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

      13. distribute-neg-in 25.4%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 65.7%

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

      16. sqr-neg 65.7%

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

      17. sqrt-unprod 83.6%

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

      18. add-sqr-sqrt 84.1%

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

      19. sub-neg 84.1%

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

   10. Applied egg-rr 84.1%

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

       1. neg-sub0 84.1%

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

       2. associate--r- 84.1%

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

       3. neg-sub0 84.1%

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

   12. Simplified 84.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-16}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 6: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6.9 \cdot 10^{+16}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -6.9e+16)
   (/ (- v) (+ t1 u))
   (if (<= t1 2.7e-15) (* (/ (- v) u) (/ t1 u)) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -6.9e+16) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 2.7e-15) {
		tmp = (-v / u) * (t1 / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-6.9d+16)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 2.7d-15) then
        tmp = (-v / u) * (t1 / u)
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -6.9e+16) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 2.7e-15) {
		tmp = (-v / u) * (t1 / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -6.9e+16:
		tmp = -v / (t1 + u)
	elif t1 <= 2.7e-15:
		tmp = (-v / u) * (t1 / u)
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -6.9e+16)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 2.7e-15)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -6.9e+16)
		tmp = -v / (t1 + u);
	elseif (t1 <= 2.7e-15)
		tmp = (-v / u) * (t1 / u);
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -6.9e+16], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.7e-15], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -6.9e16

   1. Initial program 51.0%

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

      1. associate-/r* 73.6%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 83.3%

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

      1. neg-mul-1 83.3%

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

   6. Simplified 83.3%

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

   if -6.9e16 < t1 < 2.70000000000000009e-15

   1. Initial program 83.6%

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

      1. associate-*l/ 85.8%

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

      2. *-commutative 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in t1 around 0 72.3%

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

      1. associate-*r/ 72.3%

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

      2. neg-mul-1 72.3%

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

      3. unpow2 72.3%

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

   6. Simplified 72.3%

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

   7. Taylor expanded in v around 0 73.5%

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

      1. unpow2 73.5%

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

      2. *-commutative 73.5%

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

      3. times-frac 79.8%

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

      4. associate-*r* 79.8%

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

      5. associate-*r/ 79.8%

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

      6. neg-mul-1 79.8%

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

   9. Simplified 79.8%

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

   if 2.70000000000000009e-15 < t1

   1. Initial program 70.8%

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

      1. associate-*l/ 77.0%

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

      2. *-commutative 77.0%

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

   3. Simplified 77.0%

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

      1. neg-mul-1 77.0%

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

      2. times-frac 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. associate-*l/ 98.2%

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

      2. mul-1-neg 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in t1 around inf 83.7%

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

      1. *-commutative 83.7%

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

      2. associate-*l/ 84.0%

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

      3. metadata-eval 84.0%

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

      4. neg-mul-1 84.0%

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

      5. frac-2neg 84.0%

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

      6. frac-2neg 84.0%

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

      7. remove-double-neg 84.0%

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

      8. add-sqr-sqrt 39.1%

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

      9. sqrt-unprod 42.1%

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

      10. sqr-neg 42.1%

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

      11. sqrt-unprod 11.1%

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

      12. add-sqr-sqrt 25.4%

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

      13. distribute-neg-in 25.4%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 65.7%

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

      16. sqr-neg 65.7%

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

      17. sqrt-unprod 83.6%

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

      18. add-sqr-sqrt 84.1%

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

      19. sub-neg 84.1%

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

   10. Applied egg-rr 84.1%

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

       1. neg-sub0 84.1%

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

       2. associate--r- 84.1%

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

       3. neg-sub0 84.1%

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

   12. Simplified 84.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6.9 \cdot 10^{+16}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 7: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. associate-*l/ 78.5%

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

   2. *-commutative 78.5%

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

3. Simplified 78.5%

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

   1. neg-mul-1 78.5%

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

   2. times-frac 95.8%

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

5. Applied egg-rr 95.8%

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

   1. associate-*l/ 95.9%

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

   2. mul-1-neg 95.9%

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

7. Simplified 95.9%

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

8. Final simplification 95.9%

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

Alternative 8: 67.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+32} \lor \neg \left(u \leq 4.8 \cdot 10^{+48}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.6e+32) (not (<= u 4.8e+48)))
   (* t1 (/ v (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.6e+32) || !(u <= 4.8e+48)) {
		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 <= (-5.6d+32)) .or. (.not. (u <= 4.8d+48))) 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 <= -5.6e+32) || !(u <= 4.8e+48)) {
		tmp = t1 * (v / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.6e+32) or not (u <= 4.8e+48):
		tmp = t1 * (v / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.6e+32) || !(u <= 4.8e+48))
		tmp = Float64(t1 * 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 <= -5.6e+32) || ~((u <= 4.8e+48)))
		tmp = t1 * (v / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.6e+32], N[Not[LessEqual[u, 4.8e+48]], $MachinePrecision]], N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.6e32 or 4.8000000000000002e48 < u

   1. Initial program 79.3%

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

      1. associate-*l/ 78.1%

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

      2. *-commutative 78.1%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in t1 around 0 75.5%

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

      1. mul-1-neg 75.5%

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

      2. associate-/l* 80.0%

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

      3. distribute-neg-frac 80.0%

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

      4. unpow2 80.0%

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

   6. Simplified 80.0%

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

      1. expm1-log1p-u 79.8%

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

      2. expm1-udef 68.2%

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

      3. div-inv 68.2%

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

      4. add-sqr-sqrt 27.4%

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

      5. sqrt-unprod 57.0%

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

      6. sqr-neg 57.0%

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

      7. sqrt-unprod 39.9%

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

      8. add-sqr-sqrt 67.3%

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

      9. clear-num 67.3%

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

      10. associate-/r* 67.2%

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

   8. Applied egg-rr 67.2%

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

      1. expm1-def 66.9%

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

      2. expm1-log1p 67.0%

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

      3. associate-/l/ 67.1%

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

   10. Simplified 67.1%

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

   if -5.6e32 < u < 4.8000000000000002e48

   1. Initial program 69.5%

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

      1. associate-*l/ 78.9%

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

      2. *-commutative 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in t1 around inf 72.4%

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

      1. associate-*r/ 72.4%

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

      2. neg-mul-1 72.4%

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

   6. Simplified 72.4%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+32} \lor \neg \left(u \leq 4.8 \cdot 10^{+48}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 9: 58.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6.5e+136)
   (/ -1.0 (/ u v))
   (if (<= u 1.7e+119) (/ (- v) t1) (/ v (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.5e+136) {
		tmp = -1.0 / (u / v);
	} else if (u <= 1.7e+119) {
		tmp = -v / t1;
	} else {
		tmp = v / (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 <= (-6.5d+136)) then
        tmp = (-1.0d0) / (u / v)
    else if (u <= 1.7d+119) then
        tmp = -v / t1
    else
        tmp = v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.5e+136) {
		tmp = -1.0 / (u / v);
	} else if (u <= 1.7e+119) {
		tmp = -v / t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6.5e+136:
		tmp = -1.0 / (u / v)
	elif u <= 1.7e+119:
		tmp = -v / t1
	else:
		tmp = v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6.5e+136)
		tmp = Float64(-1.0 / Float64(u / v));
	elseif (u <= 1.7e+119)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6.5e+136)
		tmp = -1.0 / (u / v);
	elseif (u <= 1.7e+119)
		tmp = -v / t1;
	else
		tmp = v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6.5e+136], N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.7e+119], N[((-v) / t1), $MachinePrecision], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -6.4999999999999998e136

   1. Initial program 74.2%

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

      1. associate-*l/ 76.8%

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

      2. *-commutative 76.8%

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

   3. Simplified 76.8%

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

      1. neg-mul-1 76.8%

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

      2. times-frac 95.7%

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

   5. Applied egg-rr 95.7%

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

      1. associate-*l/ 95.7%

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

      2. mul-1-neg 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in t1 around inf 51.1%

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

   9. Taylor expanded in t1 around 0 41.5%

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

       1. *-commutative 41.5%

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

       2. associate-*l/ 41.5%

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

       3. associate-/l* 45.3%

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

   11. Applied egg-rr 45.3%

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

   if -6.4999999999999998e136 < u < 1.70000000000000007e119

   1. Initial program 71.0%

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

      1. associate-*l/ 78.8%

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

      2. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in t1 around inf 64.9%

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

      1. associate-*r/ 64.9%

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

      2. neg-mul-1 64.9%

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

   6. Simplified 64.9%

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

   if 1.70000000000000007e119 < u

   1. Initial program 86.5%

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

      1. associate-/r* 91.7%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 52.5%

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

      1. neg-mul-1 52.5%

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

   6. Simplified 52.5%

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

      1. expm1-log1p-u 49.6%

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

      2. expm1-udef 76.3%

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

      3. add-sqr-sqrt 42.5%

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

      4. sqrt-unprod 76.0%

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

      5. sqr-neg 76.0%

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

      6. sqrt-unprod 33.8%

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

      7. add-sqr-sqrt 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. expm1-def 47.2%

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

      2. expm1-log1p 47.3%

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

   10. Simplified 47.3%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]

Alternative 10: 57.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.5 \cdot 10^{+224}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -9.5e+224) (/ v u) (if (<= u 1.2e+119) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9.5e+224) {
		tmp = v / u;
	} else if (u <= 1.2e+119) {
		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 <= (-9.5d+224)) then
        tmp = v / u
    else if (u <= 1.2d+119) 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 <= -9.5e+224) {
		tmp = v / u;
	} else if (u <= 1.2e+119) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -9.5e+224:
		tmp = v / u
	elif u <= 1.2e+119:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -9.5e+224)
		tmp = Float64(v / u);
	elseif (u <= 1.2e+119)
		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 <= -9.5e+224)
		tmp = v / u;
	elseif (u <= 1.2e+119)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -9.5e+224], N[(v / u), $MachinePrecision], If[LessEqual[u, 1.2e+119], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -9.5000000000000002e224 or 1.2e119 < u

   1. Initial program 87.0%

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

      1. associate-*l/ 82.8%

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

      2. *-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. neg-mul-1 82.8%

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

      2. times-frac 89.2%

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

   5. Applied egg-rr 89.2%

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

      1. associate-*l/ 89.3%

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

      2. mul-1-neg 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in t1 around inf 52.2%

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

      1. associate-*r/ 52.2%

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

      2. *-commutative 52.2%

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

      3. metadata-eval 52.2%

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

      4. neg-mul-1 52.2%

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

      5. add-sqr-sqrt 28.8%

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

      6. sqrt-unprod 49.1%

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

      7. sqr-neg 49.1%

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

      8. sqrt-unprod 23.4%

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

      9. add-sqr-sqrt 49.0%

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

      10. clear-num 50.2%

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

   10. Applied egg-rr 50.2%

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

   11. Taylor expanded in t1 around 0 47.6%

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

   if -9.5000000000000002e224 < u < 1.2e119

   1. Initial program 69.8%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in t1 around inf 61.6%

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

      1. associate-*r/ 61.6%

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

      2. neg-mul-1 61.6%

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

   6. Simplified 61.6%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.5 \cdot 10^{+224}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 11: 57.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.5 \cdot 10^{+216}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 2.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7.5e+216) (/ (- v) u) (if (<= u 2.6e+116) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.5e+216) {
		tmp = -v / u;
	} else if (u <= 2.6e+116) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-7.5d+216)) then
        tmp = -v / u
    else if (u <= 2.6d+116) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.5e+216) {
		tmp = -v / u;
	} else if (u <= 2.6e+116) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7.5e+216:
		tmp = -v / u
	elif u <= 2.6e+116:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7.5e+216)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 2.6e+116)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -7.5e+216)
		tmp = -v / u;
	elseif (u <= 2.6e+116)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7.5e+216], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 2.6e+116], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.4999999999999994e216

   1. Initial program 84.4%

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

      1. associate-/r* 99.9%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 50.0%

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

      1. neg-mul-1 50.0%

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

   6. Simplified 50.0%

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

   7. Taylor expanded in t1 around 0 50.0%

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

      1. neg-mul-1 50.0%

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

      2. distribute-neg-frac 50.0%

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

   9. Simplified 50.0%

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

   if -7.4999999999999994e216 < u < 2.59999999999999987e116

   1. Initial program 70.1%

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

      1. associate-*l/ 77.6%

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

      2. *-commutative 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in t1 around inf 61.9%

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

      1. associate-*r/ 61.9%

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

      2. neg-mul-1 61.9%

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

   6. Simplified 61.9%

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

   if 2.59999999999999987e116 < u

   1. Initial program 86.5%

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

      1. associate-*l/ 79.7%

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

      2. *-commutative 79.7%

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

   3. Simplified 79.7%

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

      1. neg-mul-1 79.7%

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

      2. times-frac 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. associate-*l/ 87.7%

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

      2. mul-1-neg 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in t1 around inf 52.5%

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

      1. associate-*r/ 52.5%

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

      2. *-commutative 52.5%

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

      3. metadata-eval 52.5%

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

      4. neg-mul-1 52.5%

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

      5. add-sqr-sqrt 34.3%

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

      6. sqrt-unprod 47.5%

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

      7. sqr-neg 47.5%

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

      8. sqrt-unprod 18.2%

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

      9. add-sqr-sqrt 47.3%

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

      10. clear-num 47.3%

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

   10. Applied egg-rr 47.3%

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

   11. Taylor expanded in t1 around 0 45.1%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.5 \cdot 10^{+216}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 2.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 12: 58.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+136}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{+118}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -9e+136)
   (/ -1.0 (/ u v))
   (if (<= u 2.3e+118) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9e+136) {
		tmp = -1.0 / (u / v);
	} else if (u <= 2.3e+118) {
		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 <= (-9d+136)) then
        tmp = (-1.0d0) / (u / v)
    else if (u <= 2.3d+118) 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 <= -9e+136) {
		tmp = -1.0 / (u / v);
	} else if (u <= 2.3e+118) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -9e+136:
		tmp = -1.0 / (u / v)
	elif u <= 2.3e+118:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -9e+136)
		tmp = Float64(-1.0 / Float64(u / v));
	elseif (u <= 2.3e+118)
		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 <= -9e+136)
		tmp = -1.0 / (u / v);
	elseif (u <= 2.3e+118)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -9e+136], N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.3e+118], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -8.9999999999999999e136

   1. Initial program 74.2%

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

      1. associate-*l/ 76.8%

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

      2. *-commutative 76.8%

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

   3. Simplified 76.8%

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

      1. neg-mul-1 76.8%

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

      2. times-frac 95.7%

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

   5. Applied egg-rr 95.7%

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

      1. associate-*l/ 95.7%

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

      2. mul-1-neg 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in t1 around inf 51.1%

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

   9. Taylor expanded in t1 around 0 41.5%

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

       1. *-commutative 41.5%

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

       2. associate-*l/ 41.5%

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

       3. associate-/l* 45.3%

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

   11. Applied egg-rr 45.3%

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

   if -8.9999999999999999e136 < u < 2.30000000000000016e118

   1. Initial program 71.0%

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

      1. associate-*l/ 78.8%

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

      2. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in t1 around inf 64.9%

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

      1. associate-*r/ 64.9%

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

      2. neg-mul-1 64.9%

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

   6. Simplified 64.9%

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

   if 2.30000000000000016e118 < u

   1. Initial program 86.5%

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

      1. associate-*l/ 79.7%

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

      2. *-commutative 79.7%

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

   3. Simplified 79.7%

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

      1. neg-mul-1 79.7%

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

      2. times-frac 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. associate-*l/ 87.7%

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

      2. mul-1-neg 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in t1 around inf 52.5%

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

      1. associate-*r/ 52.5%

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

      2. *-commutative 52.5%

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

      3. metadata-eval 52.5%

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

      4. neg-mul-1 52.5%

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

      5. add-sqr-sqrt 34.3%

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

      6. sqrt-unprod 47.5%

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

      7. sqr-neg 47.5%

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

      8. sqrt-unprod 18.2%

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

      9. add-sqr-sqrt 47.3%

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

      10. clear-num 47.3%

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

   10. Applied egg-rr 47.3%

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

   11. Taylor expanded in t1 around 0 45.1%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+136}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{+118}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 13: 21.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+175}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.5e+142) (/ v t1) (if (<= t1 9e+175) (/ v u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.5e+142) {
		tmp = v / t1;
	} else if (t1 <= 9e+175) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.5d+142)) then
        tmp = v / t1
    else if (t1 <= 9d+175) then
        tmp = v / u
    else
        tmp = v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.5e+142) {
		tmp = v / t1;
	} else if (t1 <= 9e+175) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.5e+142:
		tmp = v / t1
	elif t1 <= 9e+175:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.5e+142)
		tmp = Float64(v / t1);
	elseif (t1 <= 9e+175)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.5e+142)
		tmp = v / t1;
	elseif (t1 <= 9e+175)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.5e+142], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 9e+175], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.5000000000000001e142 or 8.99999999999999979e175 < t1

   1. Initial program 44.2%

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

      1. associate-*l/ 45.3%

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

      2. *-commutative 45.3%

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

   3. Simplified 45.3%

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

      1. neg-mul-1 45.3%

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

      2. times-frac 97.9%

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

   5. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. mul-1-neg 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in t1 around inf 92.3%

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

      1. associate-*r/ 92.6%

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

      2. *-commutative 92.6%

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

      3. metadata-eval 92.6%

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

      4. neg-mul-1 92.6%

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

      5. add-sqr-sqrt 47.9%

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

      6. sqrt-unprod 59.1%

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

      7. sqr-neg 59.1%

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

      8. sqrt-unprod 22.0%

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

      9. add-sqr-sqrt 42.6%

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

      10. clear-num 42.6%

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

   10. Applied egg-rr 42.6%

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

   11. Taylor expanded in t1 around inf 42.5%

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

   if -2.5000000000000001e142 < t1 < 8.99999999999999979e175

   1. Initial program 81.3%

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

      1. associate-*l/ 87.0%

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

      2. *-commutative 87.0%

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

   3. Simplified 87.0%

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

      1. neg-mul-1 87.0%

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

      2. times-frac 95.3%

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

   5. Applied egg-rr 95.3%

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

      1. associate-*l/ 95.3%

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

      2. mul-1-neg 95.3%

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

   7. Simplified 95.3%

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

   8. Taylor expanded in t1 around inf 52.6%

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

      1. associate-*r/ 52.8%

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

      2. *-commutative 52.8%

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

      3. metadata-eval 52.8%

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

      4. neg-mul-1 52.8%

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

      5. add-sqr-sqrt 23.7%

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

      6. sqrt-unprod 29.8%

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

      7. sqr-neg 29.8%

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

      8. sqrt-unprod 8.6%

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

      9. add-sqr-sqrt 18.4%

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

      10. clear-num 19.0%

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

   10. Applied egg-rr 19.0%

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

   11. Taylor expanded in t1 around 0 20.5%

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

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+175}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 14: 61.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. associate-/r* 83.9%

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

   2. associate-/l* 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in t1 around inf 60.9%

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

   1. neg-mul-1 60.9%

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

6. Simplified 60.9%

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

7. Final simplification 60.9%

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

Alternative 15: 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 73.8%

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

   1. associate-*l/ 78.5%

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

   2. *-commutative 78.5%

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

3. Simplified 78.5%

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

   1. neg-mul-1 78.5%

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

   2. times-frac 95.8%

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

5. Applied egg-rr 95.8%

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

   1. associate-*l/ 95.9%

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

   2. mul-1-neg 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in t1 around inf 60.7%

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

   1. *-commutative 60.7%

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

   2. associate-*l/ 60.9%

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

   3. metadata-eval 60.9%

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

   4. neg-mul-1 60.9%

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

   5. frac-2neg 60.9%

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

   6. frac-2neg 60.9%

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

   7. remove-double-neg 60.9%

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

   8. add-sqr-sqrt 28.6%

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

   9. sqrt-unprod 35.8%

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

   10. sqr-neg 35.8%

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

   11. sqrt-unprod 11.3%

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

   12. add-sqr-sqrt 23.3%

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

   13. distribute-neg-in 23.3%

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

   14. add-sqr-sqrt 11.7%

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

   15. sqrt-unprod 38.6%

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

   16. sqr-neg 38.6%

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

   17. sqrt-unprod 32.4%

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

   18. add-sqr-sqrt 60.6%

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

   19. sub-neg 60.6%

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

10. Applied egg-rr 60.6%

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

    1. neg-sub0 60.6%

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

    2. associate--r- 60.6%

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

    3. neg-sub0 60.6%

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

12. Simplified 60.6%

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

13. Final simplification 60.6%

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

Alternative 16: 13.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. associate-*l/ 78.5%

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

   2. *-commutative 78.5%

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

3. Simplified 78.5%

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

   1. neg-mul-1 78.5%

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

   2. times-frac 95.8%

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

5. Applied egg-rr 95.8%

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

   1. associate-*l/ 95.9%

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

   2. mul-1-neg 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in t1 around inf 60.7%

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

   1. associate-*r/ 60.9%

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

   2. *-commutative 60.9%

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

   3. metadata-eval 60.9%

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

   4. neg-mul-1 60.9%

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

   5. add-sqr-sqrt 28.6%

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

   6. sqrt-unprod 35.8%

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

   7. sqr-neg 35.8%

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

   8. sqrt-unprod 11.3%

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

   9. add-sqr-sqrt 23.3%

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

   10. clear-num 23.8%

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

10. Applied egg-rr 23.8%

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

11. Taylor expanded in t1 around inf 12.0%

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

12. Final simplification 12.0%

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

Reproduce

herbie shell --seed 2023293 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))