Rosa's DopplerBench

Percentage Accurate: 73.6% → 98.1%

Time: 10.3s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. times-frac 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{-61}:\\ \;\;\;\;-\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 9.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.7e-61)
   (- (* (/ v u) (/ t1 u)))
   (if (<= u 9.2e-28) (/ (- v) t1) (/ t1 (* (- t1 u) (/ u v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.7e-61) {
		tmp = -((v / u) * (t1 / u));
	} else if (u <= 9.2e-28) {
		tmp = -v / t1;
	} else {
		tmp = t1 / ((t1 - u) * (u / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.7d-61)) then
        tmp = -((v / u) * (t1 / u))
    else if (u <= 9.2d-28) then
        tmp = -v / t1
    else
        tmp = t1 / ((t1 - u) * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.7e-61) {
		tmp = -((v / u) * (t1 / u));
	} else if (u <= 9.2e-28) {
		tmp = -v / t1;
	} else {
		tmp = t1 / ((t1 - u) * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.7e-61:
		tmp = -((v / u) * (t1 / u))
	elif u <= 9.2e-28:
		tmp = -v / t1
	else:
		tmp = t1 / ((t1 - u) * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.7e-61)
		tmp = Float64(-Float64(Float64(v / u) * Float64(t1 / u)));
	elseif (u <= 9.2e-28)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(t1 / Float64(Float64(t1 - u) * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.7e-61)
		tmp = -((v / u) * (t1 / u));
	elseif (u <= 9.2e-28)
		tmp = -v / t1;
	else
		tmp = t1 / ((t1 - u) * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.7e-61], (-N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]), If[LessEqual[u, 9.2e-28], N[((-v) / t1), $MachinePrecision], N[(t1 / N[(N[(t1 - u), $MachinePrecision] * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.69999999999999993e-61

   1. Initial program 73.6%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 80.5%

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

   5. Taylor expanded in t1 around 0 80.0%

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

      1. associate-*r/ 80.0%

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

      2. mul-1-neg 80.0%

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

   7. Simplified 80.0%

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

   if -2.69999999999999993e-61 < u < 9.19999999999999942e-28

   1. Initial program 66.8%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

   6. Simplified 86.1%

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

   if 9.19999999999999942e-28 < u

   1. Initial program 84.5%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. clear-num 98.4%

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

      3. frac-2neg 98.4%

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

      4. frac-times 94.5%

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

      5. *-un-lft-identity 94.5%

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

      6. remove-double-neg 94.5%

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

      7. distribute-neg-in 94.5%

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

      8. add-sqr-sqrt 49.2%

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

      9. sqrt-unprod 89.4%

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

      10. sqr-neg 89.4%

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

      11. sqrt-unprod 42.7%

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

      12. add-sqr-sqrt 86.9%

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

      13. sub-neg 86.9%

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

   5. Applied egg-rr 86.9%

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

   6. Taylor expanded in t1 around 0 87.1%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{-61}:\\ \;\;\;\;-\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 9.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 3: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{v}{u}}{1 - \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5e-64)
   (/ (/ v u) (- 1.0 (/ u t1)))
   (if (<= u 6.2e-27) (/ (- v) t1) (/ t1 (* (- t1 u) (/ u v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e-64) {
		tmp = (v / u) / (1.0 - (u / t1));
	} else if (u <= 6.2e-27) {
		tmp = -v / t1;
	} else {
		tmp = t1 / ((t1 - u) * (u / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5d-64)) then
        tmp = (v / u) / (1.0d0 - (u / t1))
    else if (u <= 6.2d-27) then
        tmp = -v / t1
    else
        tmp = t1 / ((t1 - u) * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e-64) {
		tmp = (v / u) / (1.0 - (u / t1));
	} else if (u <= 6.2e-27) {
		tmp = -v / t1;
	} else {
		tmp = t1 / ((t1 - u) * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5e-64:
		tmp = (v / u) / (1.0 - (u / t1))
	elif u <= 6.2e-27:
		tmp = -v / t1
	else:
		tmp = t1 / ((t1 - u) * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5e-64)
		tmp = Float64(Float64(v / u) / Float64(1.0 - Float64(u / t1)));
	elseif (u <= 6.2e-27)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(t1 / Float64(Float64(t1 - u) * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5e-64)
		tmp = (v / u) / (1.0 - (u / t1));
	elseif (u <= 6.2e-27)
		tmp = -v / t1;
	else
		tmp = t1 / ((t1 - u) * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5e-64], N[(N[(v / u), $MachinePrecision] / N[(1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 6.2e-27], N[((-v) / t1), $MachinePrecision], N[(t1 / N[(N[(t1 - u), $MachinePrecision] * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.00000000000000033e-64

   1. Initial program 73.6%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 80.5%

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

      1. expm1-log1p-u 76.1%

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

      2. expm1-udef 53.3%

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

   6. Applied egg-rr 53.2%

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

      1. expm1-def 63.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{u \cdot \left(1 - \frac{u}{t1}\right)}\right)\right)} \]

      2. expm1-log1p 67.6%

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

      3. associate-/r* 80.3%

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

   8. Simplified 80.3%

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

   if -5.00000000000000033e-64 < u < 6.1999999999999997e-27

   1. Initial program 66.8%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

   6. Simplified 86.1%

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

   if 6.1999999999999997e-27 < u

   1. Initial program 84.5%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. clear-num 98.4%

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

      3. frac-2neg 98.4%

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

      4. frac-times 94.5%

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

      5. *-un-lft-identity 94.5%

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

      6. remove-double-neg 94.5%

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

      7. distribute-neg-in 94.5%

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

      8. add-sqr-sqrt 49.2%

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

      9. sqrt-unprod 89.4%

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

      10. sqr-neg 89.4%

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

      11. sqrt-unprod 42.7%

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

      12. add-sqr-sqrt 86.9%

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

      13. sub-neg 86.9%

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

   5. Applied egg-rr 86.9%

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

   6. Taylor expanded in t1 around 0 87.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{v}{u}}{1 - \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 4: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6.4e-60)
   (* (/ (- t1) (+ t1 u)) (/ v u))
   (if (<= u 3.1e-28) (/ (- v) t1) (/ t1 (* (- t1 u) (/ u v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.4e-60) {
		tmp = (-t1 / (t1 + u)) * (v / u);
	} else if (u <= 3.1e-28) {
		tmp = -v / t1;
	} else {
		tmp = t1 / ((t1 - u) * (u / v));
	}
	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 <= (-6.4d-60)) then
        tmp = (-t1 / (t1 + u)) * (v / u)
    else if (u <= 3.1d-28) then
        tmp = -v / t1
    else
        tmp = t1 / ((t1 - u) * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.4e-60) {
		tmp = (-t1 / (t1 + u)) * (v / u);
	} else if (u <= 3.1e-28) {
		tmp = -v / t1;
	} else {
		tmp = t1 / ((t1 - u) * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6.4e-60:
		tmp = (-t1 / (t1 + u)) * (v / u)
	elif u <= 3.1e-28:
		tmp = -v / t1
	else:
		tmp = t1 / ((t1 - u) * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6.4e-60)
		tmp = Float64(Float64(Float64(-t1) / Float64(t1 + u)) * Float64(v / u));
	elseif (u <= 3.1e-28)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(t1 / Float64(Float64(t1 - u) * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6.4e-60)
		tmp = (-t1 / (t1 + u)) * (v / u);
	elseif (u <= 3.1e-28)
		tmp = -v / t1;
	else
		tmp = t1 / ((t1 - u) * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6.4e-60], N[(N[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.1e-28], N[((-v) / t1), $MachinePrecision], N[(t1 / N[(N[(t1 - u), $MachinePrecision] * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -6.4000000000000003e-60

   1. Initial program 73.6%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 80.5%

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

   if -6.4000000000000003e-60 < u < 3.09999999999999992e-28

   1. Initial program 66.8%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

   6. Simplified 86.1%

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

   if 3.09999999999999992e-28 < u

   1. Initial program 84.5%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. clear-num 98.4%

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

      3. frac-2neg 98.4%

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

      4. frac-times 94.5%

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

      5. *-un-lft-identity 94.5%

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

      6. remove-double-neg 94.5%

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

      7. distribute-neg-in 94.5%

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

      8. add-sqr-sqrt 49.2%

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

      9. sqrt-unprod 89.4%

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

      10. sqr-neg 89.4%

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

      11. sqrt-unprod 42.7%

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

      12. add-sqr-sqrt 86.9%

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

      13. sub-neg 86.9%

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

   5. Applied egg-rr 86.9%

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

   6. Taylor expanded in t1 around 0 87.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 5: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{\frac{t1 - u}{t1}}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{-28}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7.2e-72)
   (/ (/ v (+ t1 u)) (/ (- t1 u) t1))
   (if (<= u 3.3e-28) (/ (- v) t1) (/ t1 (* (- t1 u) (/ u v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.2e-72) {
		tmp = (v / (t1 + u)) / ((t1 - u) / t1);
	} else if (u <= 3.3e-28) {
		tmp = -v / t1;
	} else {
		tmp = t1 / ((t1 - u) * (u / v));
	}
	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 <= (-7.2d-72)) then
        tmp = (v / (t1 + u)) / ((t1 - u) / t1)
    else if (u <= 3.3d-28) then
        tmp = -v / t1
    else
        tmp = t1 / ((t1 - u) * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.2e-72) {
		tmp = (v / (t1 + u)) / ((t1 - u) / t1);
	} else if (u <= 3.3e-28) {
		tmp = -v / t1;
	} else {
		tmp = t1 / ((t1 - u) * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7.2e-72:
		tmp = (v / (t1 + u)) / ((t1 - u) / t1)
	elif u <= 3.3e-28:
		tmp = -v / t1
	else:
		tmp = t1 / ((t1 - u) * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7.2e-72)
		tmp = Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(t1 - u) / t1));
	elseif (u <= 3.3e-28)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(t1 / Float64(Float64(t1 - u) * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -7.2e-72)
		tmp = (v / (t1 + u)) / ((t1 - u) / t1);
	elseif (u <= 3.3e-28)
		tmp = -v / t1;
	else
		tmp = t1 / ((t1 - u) * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7.2e-72], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(N[(t1 - u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.3e-28], N[((-v) / t1), $MachinePrecision], N[(t1 / N[(N[(t1 - u), $MachinePrecision] * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.2e-72

   1. Initial program 73.6%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 87.0%

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

      3. *-un-lft-identity 87.0%

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

      4. frac-2neg 87.0%

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

      5. distribute-neg-in 87.0%

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

      6. add-sqr-sqrt 43.3%

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

      7. sqrt-unprod 73.8%

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

      8. sqr-neg 73.8%

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

      9. sqrt-unprod 36.7%

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

      10. add-sqr-sqrt 71.3%

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

      11. sub-neg 71.3%

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

      12. remove-double-neg 71.3%

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

   5. Applied egg-rr 71.3%

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

      1. associate-/l/ 84.0%

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

   7. Simplified 84.0%

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

   if -7.2e-72 < u < 3.3000000000000002e-28

   1. Initial program 66.8%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

   6. Simplified 86.1%

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

   if 3.3000000000000002e-28 < u

   1. Initial program 84.5%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. clear-num 98.4%

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

      3. frac-2neg 98.4%

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

      4. frac-times 94.5%

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

      5. *-un-lft-identity 94.5%

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

      6. remove-double-neg 94.5%

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

      7. distribute-neg-in 94.5%

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

      8. add-sqr-sqrt 49.2%

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

      9. sqrt-unprod 89.4%

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

      10. sqr-neg 89.4%

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

      11. sqrt-unprod 42.7%

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

      12. add-sqr-sqrt 86.9%

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

      13. sub-neg 86.9%

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

   5. Applied egg-rr 86.9%

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

   6. Taylor expanded in t1 around 0 87.1%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{\frac{t1 - u}{t1}}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{-28}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 6: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 2.7 \cdot 10^{-52}:\\ \;\;\;\;-\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.5e-50)
   (/ v (- u t1))
   (if (<= t1 2.7e-52) (- (* (/ v u) (/ t1 u))) (/ v (- (* u -2.0) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.5e-50) {
		tmp = v / (u - t1);
	} else if (t1 <= 2.7e-52) {
		tmp = -((v / u) * (t1 / u));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.5d-50)) then
        tmp = v / (u - t1)
    else if (t1 <= 2.7d-52) then
        tmp = -((v / u) * (t1 / u))
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.5e-50) {
		tmp = v / (u - t1);
	} else if (t1 <= 2.7e-52) {
		tmp = -((v / u) * (t1 / u));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.5e-50:
		tmp = v / (u - t1)
	elif t1 <= 2.7e-52:
		tmp = -((v / u) * (t1 / u))
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.5e-50)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= 2.7e-52)
		tmp = Float64(-Float64(Float64(v / u) * Float64(t1 / u)));
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.5e-50)
		tmp = v / (u - t1);
	elseif (t1 <= 2.7e-52)
		tmp = -((v / u) * (t1 / u));
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.5e-50], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.7e-52], (-N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]), N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.49999999999999984e-50

   1. Initial program 60.2%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. frac-2neg 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. add-sqr-sqrt 99.3%

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

      8. sqrt-unprod 68.0%

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

      9. sqr-neg 68.0%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 39.8%

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

      12. sub-neg 39.8%

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

      13. remove-double-neg 39.8%

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

   5. Applied egg-rr 39.8%

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

   6. Taylor expanded in t1 around inf 31.1%

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

      1. clear-num 31.2%

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

      2. associate-/r/ 31.1%

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

      3. frac-2neg 31.1%

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

      4. metadata-eval 31.1%

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

      5. distribute-neg-in 31.1%

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

      6. add-sqr-sqrt 31.1%

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

      7. sqrt-unprod 31.7%

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

      8. sqr-neg 31.7%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 89.7%

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

      11. sub-neg 89.7%

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

   8. Applied egg-rr 89.7%

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

      1. expm1-log1p-u 83.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{t1 - u} \cdot v\right)\right)} \]

      2. expm1-udef 37.5%

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

      3. *-commutative 37.5%

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

      4. frac-2neg 37.5%

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

      5. metadata-eval 37.5%

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

      6. un-div-inv 37.5%

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

      7. sub-neg 37.5%

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

      8. distribute-neg-in 37.5%

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

      9. remove-double-neg 37.5%

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

   10. Applied egg-rr 37.5%

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

       1. expm1-def 83.6%

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

       2. expm1-log1p 90.0%

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

       3. +-commutative 90.0%

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

       4. unsub-neg 90.0%

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

   12. Simplified 90.0%

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

   if -2.49999999999999984e-50 < t1 < 2.70000000000000009e-52

   1. Initial program 81.5%

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

      1. times-frac 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in t1 around 0 80.0%

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

   5. Taylor expanded in t1 around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. mul-1-neg 81.9%

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

   7. Simplified 81.9%

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

   if 2.70000000000000009e-52 < t1

   1. Initial program 74.1%

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

      1. associate-/r* 86.4%

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

      2. *-commutative 86.4%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 90.9%

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

      5. +-commutative 90.9%

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

      6. remove-double-neg 90.9%

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

      7. unsub-neg 90.9%

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

      8. div-sub 90.9%

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

      9. sub-neg 90.9%

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

      10. *-inverses 90.9%

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

      11. metadata-eval 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in t1 around inf 78.6%

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

      1. mul-1-neg 78.6%

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

      2. unsub-neg 78.6%

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

      3. *-commutative 78.6%

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

   6. Simplified 78.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 2.7 \cdot 10^{-52}:\\ \;\;\;\;-\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 7: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{-61}:\\ \;\;\;\;-\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.2e-61)
   (- (* (/ v u) (/ t1 u)))
   (if (<= u 3.8e-30) (/ (- v) t1) (/ t1 (* u (/ (- u) v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.2e-61) {
		tmp = -((v / u) * (t1 / u));
	} else if (u <= 3.8e-30) {
		tmp = -v / t1;
	} else {
		tmp = t1 / (u * (-u / v));
	}
	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 <= (-5.2d-61)) then
        tmp = -((v / u) * (t1 / u))
    else if (u <= 3.8d-30) then
        tmp = -v / t1
    else
        tmp = t1 / (u * (-u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.2e-61) {
		tmp = -((v / u) * (t1 / u));
	} else if (u <= 3.8e-30) {
		tmp = -v / t1;
	} else {
		tmp = t1 / (u * (-u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.2e-61:
		tmp = -((v / u) * (t1 / u))
	elif u <= 3.8e-30:
		tmp = -v / t1
	else:
		tmp = t1 / (u * (-u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.2e-61)
		tmp = Float64(-Float64(Float64(v / u) * Float64(t1 / u)));
	elseif (u <= 3.8e-30)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(-u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.2e-61)
		tmp = -((v / u) * (t1 / u));
	elseif (u <= 3.8e-30)
		tmp = -v / t1;
	else
		tmp = t1 / (u * (-u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.2e-61], (-N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]), If[LessEqual[u, 3.8e-30], N[((-v) / t1), $MachinePrecision], N[(t1 / N[(u * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.20000000000000021e-61

   1. Initial program 73.6%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 80.5%

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

   5. Taylor expanded in t1 around 0 80.0%

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

      1. associate-*r/ 80.0%

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

      2. mul-1-neg 80.0%

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

   7. Simplified 80.0%

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

   if -5.20000000000000021e-61 < u < 3.8000000000000003e-30

   1. Initial program 66.8%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

   6. Simplified 86.1%

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

   if 3.8000000000000003e-30 < u

   1. Initial program 84.5%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around 0 80.9%

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

   5. Taylor expanded in t1 around 0 80.1%

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

      1. associate-*r/ 80.1%

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

      2. mul-1-neg 80.1%

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

   7. Simplified 80.1%

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

      1. *-commutative 80.1%

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

      2. clear-num 81.3%

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

      3. frac-2neg 81.3%

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

      4. frac-times 83.6%

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

      5. *-un-lft-identity 83.6%

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

      6. remove-double-neg 83.6%

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

   9. Applied egg-rr 83.6%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{-61}:\\ \;\;\;\;-\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \end{array} \]

Alternative 8: 67.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.6 \cdot 10^{+104} \lor \neg \left(u \leq 2.2 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.6e+104) (not (<= u 2.2e+86)))
   (/ v (* u (/ u t1)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.6e+104) || !(u <= 2.2e+86)) {
		tmp = v / (u * (u / 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 <= (-6.6d+104)) .or. (.not. (u <= 2.2d+86))) then
        tmp = v / (u * (u / 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 <= -6.6e+104) || !(u <= 2.2e+86)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.6e+104) or not (u <= 2.2e+86):
		tmp = v / (u * (u / t1))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.6e+104) || !(u <= 2.2e+86))
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.6e+104) || ~((u <= 2.2e+86)))
		tmp = v / (u * (u / t1));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.6e+104], N[Not[LessEqual[u, 2.2e+86]], $MachinePrecision]], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.59999999999999969e104 or 2.20000000000000003e86 < u

   1. Initial program 76.4%

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

      1. times-frac 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in t1 around 0 93.0%

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

   5. Taylor expanded in t1 around 0 92.8%

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

      1. associate-*r/ 92.8%

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

      2. mul-1-neg 92.8%

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

   7. Simplified 92.8%

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

      1. clear-num 92.8%

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

      2. frac-times 74.5%

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

      3. *-un-lft-identity 74.5%

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

      4. add-sqr-sqrt 40.4%

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

      5. sqrt-unprod 61.2%

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

      6. sqr-neg 61.2%

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

      7. sqrt-unprod 29.7%

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

      8. add-sqr-sqrt 68.0%

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

   9. Applied egg-rr 68.0%

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

   if -6.59999999999999969e104 < u < 2.20000000000000003e86

   1. Initial program 72.6%

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

      1. times-frac 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r/ 98.8%

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

      2. clear-num 98.8%

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

      3. associate-*l/ 98.7%

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

      4. *-un-lft-identity 98.7%

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

      5. frac-2neg 98.7%

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

      6. distribute-neg-in 98.7%

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

      7. add-sqr-sqrt 41.5%

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

      8. sqrt-unprod 57.4%

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

      9. sqr-neg 57.4%

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

      10. sqrt-unprod 27.1%

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

      11. add-sqr-sqrt 43.2%

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

      12. sub-neg 43.2%

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

      13. remove-double-neg 43.2%

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

   5. Applied egg-rr 43.2%

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

      1. clear-num 43.1%

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

      2. associate-/r/ 43.2%

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

      3. clear-num 43.2%

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

      4. add-sqr-sqrt 27.1%

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

      5. sqrt-unprod 31.2%

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

      6. sqr-neg 31.2%

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

      7. sqrt-unprod 33.9%

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

      8. add-sqr-sqrt 78.1%

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

      9. sub-neg 78.1%

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

      10. add-sqr-sqrt 43.8%

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

      11. sqrt-unprod 48.1%

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

      12. sqr-neg 48.1%

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

      13. sqrt-unprod 9.8%

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

      14. add-sqr-sqrt 24.3%

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

      15. distribute-neg-in 24.3%

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

      16. frac-2neg 24.3%

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

      17. associate-/r/ 24.4%

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

      18. frac-2neg 24.4%

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

      19. associate-/r/ 24.3%

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

      20. remove-double-neg 24.3%

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

      21. frac-2neg 24.3%

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

   7. Applied egg-rr 98.8%

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

   8. Taylor expanded in t1 around inf 75.1%

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

      1. mul-1-neg 75.1%

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

   10. Simplified 75.1%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.6 \cdot 10^{+104} \lor \neg \left(u \leq 2.2 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 9: 68.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+105} \lor \neg \left(u \leq 2.35 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.55e+105) (not (<= u 2.35e+86)))
   (/ v (* u (/ u t1)))
   (/ v (- (* u -2.0) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.55e+105) || !(u <= 2.35e+86)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.55d+105)) .or. (.not. (u <= 2.35d+86))) then
        tmp = v / (u * (u / t1))
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.55e+105) || !(u <= 2.35e+86)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.55e+105) or not (u <= 2.35e+86):
		tmp = v / (u * (u / t1))
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.55e+105) || !(u <= 2.35e+86))
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.55e+105) || ~((u <= 2.35e+86)))
		tmp = v / (u * (u / t1));
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.55e+105], N[Not[LessEqual[u, 2.35e+86]], $MachinePrecision]], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.55000000000000002e105 or 2.3500000000000001e86 < u

   1. Initial program 76.4%

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

      1. times-frac 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in t1 around 0 93.0%

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

   5. Taylor expanded in t1 around 0 92.8%

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

      1. associate-*r/ 92.8%

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

      2. mul-1-neg 92.8%

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

   7. Simplified 92.8%

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

      1. clear-num 92.8%

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

      2. frac-times 74.5%

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

      3. *-un-lft-identity 74.5%

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

      4. add-sqr-sqrt 40.4%

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

      5. sqrt-unprod 61.2%

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

      6. sqr-neg 61.2%

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

      7. sqrt-unprod 29.7%

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

      8. add-sqr-sqrt 68.0%

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

   9. Applied egg-rr 68.0%

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

   if -1.55000000000000002e105 < u < 2.3500000000000001e86

   1. Initial program 72.6%

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

      1. associate-/r* 83.5%

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

      2. *-commutative 83.5%

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

      3. associate-/l* 98.7%

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

      4. associate-/l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. div-sub 99.9%

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

      9. sub-neg 99.9%

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

      10. *-inverses 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. unsub-neg 75.3%

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

      3. *-commutative 75.3%

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

   6. Simplified 75.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+105} \lor \neg \left(u \leq 2.35 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 10: 67.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+57}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{u}\\ \mathbf{elif}\;u \leq 1.85 \cdot 10^{+86}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.6e+57)
   (/ (* t1 (/ v u)) u)
   (if (<= u 1.85e+86) (/ v (- (* u -2.0) t1)) (/ v (* u (/ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.6e+57) {
		tmp = (t1 * (v / u)) / u;
	} else if (u <= 1.85e+86) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.6d+57)) then
        tmp = (t1 * (v / u)) / u
    else if (u <= 1.85d+86) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = v / (u * (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.6e+57) {
		tmp = (t1 * (v / u)) / u;
	} else if (u <= 1.85e+86) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.6e+57:
		tmp = (t1 * (v / u)) / u
	elif u <= 1.85e+86:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = v / (u * (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.6e+57)
		tmp = Float64(Float64(t1 * Float64(v / u)) / u);
	elseif (u <= 1.85e+86)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.6e+57)
		tmp = (t1 * (v / u)) / u;
	elseif (u <= 1.85e+86)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = v / (u * (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.6e+57], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 1.85e+86], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.6000000000000002e57

   1. Initial program 74.2%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 91.1%

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

   5. Taylor expanded in t1 around 0 90.6%

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

      1. associate-*r/ 90.6%

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

      2. mul-1-neg 90.6%

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

   7. Simplified 90.6%

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

      1. associate-*l/ 90.6%

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

      2. add-sqr-sqrt 41.3%

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

      3. sqrt-unprod 67.4%

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

      4. sqr-neg 67.4%

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

      5. sqrt-unprod 32.4%

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

      6. add-sqr-sqrt 64.4%

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

   9. Applied egg-rr 64.4%

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

   if -3.6000000000000002e57 < u < 1.84999999999999996e86

   1. Initial program 72.3%

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

      1. associate-/r* 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-/l* 98.7%

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

      4. associate-/l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. div-sub 99.9%

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

      9. sub-neg 99.9%

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

      10. *-inverses 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. unsub-neg 75.3%

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

      3. *-commutative 75.3%

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

   6. Simplified 75.3%

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

   if 1.84999999999999996e86 < u

   1. Initial program 78.9%

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

      1. times-frac 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in t1 around 0 92.5%

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

   5. Taylor expanded in t1 around 0 92.6%

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

      1. associate-*r/ 92.6%

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

      2. mul-1-neg 92.6%

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

   7. Simplified 92.6%

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

      1. clear-num 92.5%

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

      2. frac-times 77.2%

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

      3. *-un-lft-identity 77.2%

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

      4. add-sqr-sqrt 45.9%

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

      5. sqrt-unprod 63.1%

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

      6. sqr-neg 63.1%

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

      7. sqrt-unprod 27.4%

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

      8. add-sqr-sqrt 73.1%

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

   9. Applied egg-rr 73.1%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+57}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{u}\\ \mathbf{elif}\;u \leq 1.85 \cdot 10^{+86}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \]

Alternative 11: 67.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{u}{t1}}\\ \mathbf{elif}\;u \leq 2.35 \cdot 10^{+86}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.8e+55)
   (/ (/ v u) (/ u t1))
   (if (<= u 2.35e+86) (/ v (- (* u -2.0) t1)) (/ v (* u (/ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.8e+55) {
		tmp = (v / u) / (u / t1);
	} else if (u <= 2.35e+86) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.8d+55)) then
        tmp = (v / u) / (u / t1)
    else if (u <= 2.35d+86) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = v / (u * (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.8e+55) {
		tmp = (v / u) / (u / t1);
	} else if (u <= 2.35e+86) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.8e+55:
		tmp = (v / u) / (u / t1)
	elif u <= 2.35e+86:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = v / (u * (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.8e+55)
		tmp = Float64(Float64(v / u) / Float64(u / t1));
	elseif (u <= 2.35e+86)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.8e+55)
		tmp = (v / u) / (u / t1);
	elseif (u <= 2.35e+86)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = v / (u * (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.8e+55], N[(N[(v / u), $MachinePrecision] / N[(u / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.35e+86], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.7999999999999998e55

   1. Initial program 74.2%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 91.1%

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

   5. Taylor expanded in t1 around 0 90.6%

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

      1. associate-*r/ 90.6%

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

      2. mul-1-neg 90.6%

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

   7. Simplified 90.6%

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

      1. *-commutative 90.6%

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

      2. clear-num 90.5%

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

      3. un-div-inv 90.5%

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

      4. add-sqr-sqrt 41.3%

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

      5. sqrt-unprod 67.4%

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

      6. sqr-neg 67.4%

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

      7. sqrt-unprod 32.4%

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

      8. add-sqr-sqrt 64.5%

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

   9. Applied egg-rr 64.5%

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

   if -4.7999999999999998e55 < u < 2.3500000000000001e86

   1. Initial program 72.3%

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

      1. associate-/r* 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-/l* 98.7%

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

      4. associate-/l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. div-sub 99.9%

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

      9. sub-neg 99.9%

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

      10. *-inverses 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. unsub-neg 75.3%

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

      3. *-commutative 75.3%

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

   6. Simplified 75.3%

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

   if 2.3500000000000001e86 < u

   1. Initial program 78.9%

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

      1. times-frac 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in t1 around 0 92.5%

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

   5. Taylor expanded in t1 around 0 92.6%

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

      1. associate-*r/ 92.6%

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

      2. mul-1-neg 92.6%

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

   7. Simplified 92.6%

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

      1. clear-num 92.5%

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

      2. frac-times 77.2%

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

      3. *-un-lft-identity 77.2%

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

      4. add-sqr-sqrt 45.9%

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

      5. sqrt-unprod 63.1%

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

      6. sqr-neg 63.1%

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

      7. sqrt-unprod 27.4%

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

      8. add-sqr-sqrt 73.1%

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

   9. Applied egg-rr 73.1%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{u}{t1}}\\ \mathbf{elif}\;u \leq 2.35 \cdot 10^{+86}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \]

Alternative 12: 57.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+58} \lor \neg \left(u \leq 1.1 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.2e+58) (not (<= u 1.1e+138))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+58) || !(u <= 1.1e+138)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.2d+58)) .or. (.not. (u <= 1.1d+138))) then
        tmp = v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+58) || !(u <= 1.1e+138)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.2e+58) or not (u <= 1.1e+138):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.2e+58) || !(u <= 1.1e+138))
		tmp = Float64(v / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.2e+58) || ~((u <= 1.1e+138)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.2e+58], N[Not[LessEqual[u, 1.1e+138]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.2e58 or 1.1e138 < u

   1. Initial program 76.9%

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

      1. times-frac 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in t1 around 0 94.0%

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

      1. expm1-log1p-u 91.8%

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

      2. expm1-udef 71.8%

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

   6. Applied egg-rr 71.8%

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

      1. expm1-def 74.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{u \cdot \left(1 - \frac{u}{t1}\right)}\right)\right)} \]

      2. expm1-log1p 76.3%

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

      3. associate-/r* 93.9%

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

   8. Simplified 93.9%

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

   9. Taylor expanded in u around 0 38.2%

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

   if -1.2e58 < u < 1.1e138

   1. Initial program 72.4%

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

      1. times-frac 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t1 around inf 71.8%

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

      1. associate-*r/ 71.8%

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

      2. neg-mul-1 71.8%

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

   6. Simplified 71.8%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+58} \lor \neg \left(u \leq 1.1 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 13: 57.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 4 \cdot 10^{+138}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.2e+58) (/ v u) (if (<= u 4e+138) (/ (- v) t1) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.2e+58) {
		tmp = v / u;
	} else if (u <= 4e+138) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.2d+58)) then
        tmp = v / u
    else if (u <= 4d+138) then
        tmp = -v / t1
    else
        tmp = -v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.2e+58) {
		tmp = v / u;
	} else if (u <= 4e+138) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.2e+58:
		tmp = v / u
	elif u <= 4e+138:
		tmp = -v / t1
	else:
		tmp = -v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.2e+58)
		tmp = Float64(v / u);
	elseif (u <= 4e+138)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-v) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.2e+58)
		tmp = v / u;
	elseif (u <= 4e+138)
		tmp = -v / t1;
	else
		tmp = -v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.2e+58], N[(v / u), $MachinePrecision], If[LessEqual[u, 4e+138], N[((-v) / t1), $MachinePrecision], N[((-v) / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.2e58

   1. Initial program 74.2%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 91.1%

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

      1. expm1-log1p-u 87.2%

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

      2. expm1-udef 67.1%

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

   6. Applied egg-rr 67.0%

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

      1. expm1-def 69.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{u \cdot \left(1 - \frac{u}{t1}\right)}\right)\right)} \]

      2. expm1-log1p 73.0%

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

      3. associate-/r* 91.0%

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

   8. Simplified 91.0%

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

   9. Taylor expanded in u around 0 35.3%

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

   if -1.2e58 < u < 4.0000000000000001e138

   1. Initial program 72.4%

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

      1. times-frac 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t1 around inf 71.8%

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

      1. associate-*r/ 71.8%

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

      2. neg-mul-1 71.8%

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

   6. Simplified 71.8%

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

   if 4.0000000000000001e138 < u

   1. Initial program 80.5%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around 0 98.0%

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

   5. Taylor expanded in t1 around inf 42.2%

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

      1. associate-*r/ 42.2%

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

      2. neg-mul-1 42.2%

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

   7. Simplified 42.2%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 4 \cdot 10^{+138}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]

Alternative 14: 57.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+25}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{+138}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4e+25)
   (/ v (+ t1 u))
   (if (<= u 1.4e+138) (/ (- v) t1) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4e+25) {
		tmp = v / (t1 + u);
	} else if (u <= 1.4e+138) {
		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 <= (-4d+25)) then
        tmp = v / (t1 + u)
    else if (u <= 1.4d+138) 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 <= -4e+25) {
		tmp = v / (t1 + u);
	} else if (u <= 1.4e+138) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4e+25:
		tmp = v / (t1 + u)
	elif u <= 1.4e+138:
		tmp = -v / t1
	else:
		tmp = -v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4e+25)
		tmp = Float64(v / Float64(t1 + u));
	elseif (u <= 1.4e+138)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-v) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4e+25)
		tmp = v / (t1 + u);
	elseif (u <= 1.4e+138)
		tmp = -v / t1;
	else
		tmp = -v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4e+25], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.4e+138], N[((-v) / t1), $MachinePrecision], N[((-v) / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.00000000000000036e25

   1. Initial program 75.1%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. frac-2neg 99.8%

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

      6. distribute-neg-in 99.8%

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

      7. add-sqr-sqrt 45.2%

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

      8. sqrt-unprod 87.8%

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

      9. sqr-neg 87.8%

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

      10. sqrt-unprod 51.0%

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

      11. add-sqr-sqrt 92.9%

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

      12. sub-neg 92.9%

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

      13. remove-double-neg 92.9%

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

   5. Applied egg-rr 92.9%

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

   6. Taylor expanded in t1 around inf 37.5%

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

   if -4.00000000000000036e25 < u < 1.4e138

   1. Initial program 72.0%

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

      1. times-frac 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t1 around inf 72.1%

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

      1. associate-*r/ 72.1%

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

      2. neg-mul-1 72.1%

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

   6. Simplified 72.1%

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

   if 1.4e138 < u

   1. Initial program 80.5%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around 0 98.0%

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

   5. Taylor expanded in t1 around inf 42.2%

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

      1. associate-*r/ 42.2%

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

      2. neg-mul-1 42.2%

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

   7. Simplified 42.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+25}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{+138}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]

Alternative 15: 23.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+81} \lor \neg \left(t1 \leq 1.9 \cdot 10^{+108}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.8e+81) (not (<= t1 1.9e+108))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.8e+81) || !(t1 <= 1.9e+108)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.8d+81)) .or. (.not. (t1 <= 1.9d+108))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.8e+81) || !(t1 <= 1.9e+108)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.8e+81) or not (t1 <= 1.9e+108):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.8e+81) || !(t1 <= 1.9e+108))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.8e+81) || ~((t1 <= 1.9e+108)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.8e+81], N[Not[LessEqual[t1, 1.9e+108]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.80000000000000003e81 or 1.90000000000000004e108 < t1

   1. Initial program 55.4%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 98.2%

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

      3. frac-2neg 98.2%

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

      4. frac-times 65.8%

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

      5. *-un-lft-identity 65.8%

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

      6. remove-double-neg 65.8%

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

      7. distribute-neg-in 65.8%

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

      8. add-sqr-sqrt 33.4%

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

      9. sqrt-unprod 50.4%

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

      10. sqr-neg 50.4%

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

      11. sqrt-unprod 26.5%

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

      12. add-sqr-sqrt 50.2%

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

      13. sub-neg 50.2%

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

   5. Applied egg-rr 50.2%

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

   6. Taylor expanded in t1 around inf 43.5%

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

   if -1.80000000000000003e81 < t1 < 1.90000000000000004e108

   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. times-frac 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around 0 66.1%

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

      1. expm1-log1p-u 62.2%

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

      2. expm1-udef 44.9%

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

   6. Applied egg-rr 45.2%

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

      1. expm1-def 55.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{u \cdot \left(1 - \frac{u}{t1}\right)}\right)\right)} \]

      2. expm1-log1p 59.5%

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

      3. associate-/r* 66.4%

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

   8. Simplified 66.4%

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

   9. Taylor expanded in u around 0 18.3%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+81} \lor \neg \left(t1 \leq 1.9 \cdot 10^{+108}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 16: 61.2% 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 73.9%

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

   1. times-frac 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r/ 98.9%

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

   2. clear-num 98.9%

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

   3. associate-*l/ 98.8%

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

   4. *-un-lft-identity 98.8%

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

   5. frac-2neg 98.8%

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

   6. distribute-neg-in 98.8%

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

   7. add-sqr-sqrt 45.9%

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

   8. sqrt-unprod 69.0%

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

   9. sqr-neg 69.0%

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

   10. sqrt-unprod 33.2%

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

   11. add-sqr-sqrt 61.2%

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

   12. sub-neg 61.2%

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

   13. remove-double-neg 61.2%

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

5. Applied egg-rr 61.2%

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

   1. clear-num 61.2%

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

   2. associate-/r/ 61.2%

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

   3. clear-num 61.3%

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

   4. add-sqr-sqrt 33.1%

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

   5. sqrt-unprod 44.7%

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

   6. sqr-neg 44.7%

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

   7. sqrt-unprod 36.9%

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

   8. add-sqr-sqrt 76.6%

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

   9. sub-neg 76.6%

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

   10. add-sqr-sqrt 39.4%

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

   11. sqrt-unprod 55.5%

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

   12. sqr-neg 55.5%

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

   13. sqrt-unprod 19.9%

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

   14. add-sqr-sqrt 40.2%

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

   15. distribute-neg-in 40.2%

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

   16. frac-2neg 40.2%

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

   17. associate-/r/ 40.2%

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

   18. frac-2neg 40.2%

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

   19. associate-/r/ 40.2%

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

   20. remove-double-neg 40.2%

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

   21. frac-2neg 40.2%

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in t1 around inf 62.5%

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

   1. mul-1-neg 62.5%

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

10. Simplified 62.5%

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

11. Final simplification 62.5%

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

Alternative 17: 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 73.9%

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

   1. times-frac 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r/ 98.9%

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

   2. clear-num 98.9%

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

   3. associate-*l/ 98.8%

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

   4. *-un-lft-identity 98.8%

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

   5. frac-2neg 98.8%

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

   6. distribute-neg-in 98.8%

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

   7. add-sqr-sqrt 45.9%

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

   8. sqrt-unprod 69.0%

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

   9. sqr-neg 69.0%

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

   10. sqrt-unprod 33.2%

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

   11. add-sqr-sqrt 61.2%

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

   12. sub-neg 61.2%

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

   13. remove-double-neg 61.2%

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

5. Applied egg-rr 61.2%

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

6. Taylor expanded in t1 around inf 25.7%

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

   1. clear-num 26.1%

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

   2. associate-/r/ 25.7%

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

   3. frac-2neg 25.7%

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

   4. metadata-eval 25.7%

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

   5. distribute-neg-in 25.7%

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

   6. add-sqr-sqrt 11.7%

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

   7. sqrt-unprod 41.4%

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

   8. sqr-neg 41.4%

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

   9. sqrt-unprod 33.1%

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

   10. add-sqr-sqrt 62.0%

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

   11. sub-neg 62.0%

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

8. Applied egg-rr 62.0%

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

   1. expm1-log1p-u 53.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{t1 - u} \cdot v\right)\right)} \]

   2. expm1-udef 44.4%

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

   3. *-commutative 44.4%

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

   4. frac-2neg 44.4%

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

   5. metadata-eval 44.4%

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

   6. un-div-inv 44.4%

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

   7. sub-neg 44.4%

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

   8. distribute-neg-in 44.4%

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

   9. remove-double-neg 44.4%

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

10. Applied egg-rr 44.4%

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

    1. expm1-def 53.2%

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

    2. expm1-log1p 62.2%

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

    3. +-commutative 62.2%

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

    4. unsub-neg 62.2%

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

12. Simplified 62.2%

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

13. Final simplification 62.2%

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

Alternative 18: 14.3% 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 73.9%

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

   1. times-frac 99.2%

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

3. Simplified 99.2%

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

   1. *-commutative 99.2%

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

   2. clear-num 98.3%

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

   3. frac-2neg 98.3%

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

   4. frac-times 83.0%

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

   5. *-un-lft-identity 83.0%

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

   6. remove-double-neg 83.0%

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

   7. distribute-neg-in 83.0%

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

   8. add-sqr-sqrt 37.9%

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

   9. sqrt-unprod 64.8%

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

   10. sqr-neg 64.8%

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

   11. sqrt-unprod 30.3%

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

   12. add-sqr-sqrt 58.0%

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

   13. sub-neg 58.0%

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

5. Applied egg-rr 58.0%

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

6. Taylor expanded in t1 around inf 15.7%

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

7. Final simplification 15.7%

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

Reproduce

herbie shell --seed 2023315 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))