Rosa's DopplerBench

Percentage Accurate: 73.1% → 97.9%

Time: 8.5s

Alternatives: 12

Speedup: 1.1×

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: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

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 71.6%

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

   1. times-frac 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 2: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 3.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 3.8e+155)
   (/ v (* (+ t1 u) (- -1.0 (/ u t1))))
   (* (/ v (+ t1 u)) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.8e+155) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 3.8d+155) then
        tmp = v / ((t1 + u) * ((-1.0d0) - (u / t1)))
    else
        tmp = (v / (t1 + u)) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.8e+155) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 3.8e+155:
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)))
	else:
		tmp = (v / (t1 + u)) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 3.8e+155)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(-1.0 - Float64(u / t1))));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 3.8e+155)
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	else
		tmp = (v / (t1 + u)) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 3.8e+155], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 3.8000000000000001e155

   1. Initial program 69.9%

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

      1. *-commutative 69.9%

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

      2. times-frac 97.8%

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

      3. neg-mul-1 97.8%

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

      4. associate-/l* 97.8%

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

      5. associate-*r/ 97.8%

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

      6. associate-/l* 97.8%

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

      7. associate-/l/ 97.8%

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

      8. neg-mul-1 97.8%

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

      9. *-lft-identity 97.8%

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

      10. metadata-eval 97.8%

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

      11. times-frac 97.8%

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

      12. neg-mul-1 97.8%

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

      13. remove-double-neg 97.8%

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

      14. neg-mul-1 97.8%

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

      15. sub0-neg 97.8%

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

      16. associate--r+ 97.8%

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

      17. neg-sub0 97.8%

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

      18. div-sub 97.8%

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

      19. distribute-frac-neg 97.8%

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

      20. *-inverses 97.8%

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

      21. metadata-eval 97.8%

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

   3. Simplified 97.8%

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

      1. expm1-log1p-u 84.0%

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

      2. expm1-udef 44.5%

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

      3. associate-/l/ 44.9%

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

   5. Applied egg-rr 44.9%

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

      1. expm1-def 82.2%

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

      2. expm1-log1p 97.3%

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

      3. *-commutative 97.3%

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

   7. Simplified 97.3%

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

   if 3.8000000000000001e155 < u

   1. Initial program 81.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-neg-frac 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 3.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 3: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -90000 \lor \neg \left(t1 \leq 7.1 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -90000.0) (not (<= t1 7.1e-62)))
   (/ v (- (* u -2.0) t1))
   (* (- t1) (/ v (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -90000.0) || !(t1 <= 7.1e-62)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = -t1 * (v / (u * 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 <= (-90000.0d0)) .or. (.not. (t1 <= 7.1d-62))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = -t1 * (v / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -90000.0) || !(t1 <= 7.1e-62)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -90000.0) or not (t1 <= 7.1e-62):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = -t1 * (v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -90000.0) || !(t1 <= 7.1e-62))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -90000.0) || ~((t1 <= 7.1e-62)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = -t1 * (v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -90000.0], N[Not[LessEqual[t1, 7.1e-62]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9e4 or 7.1000000000000001e-62 < t1

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 90.8%

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

      2. expm1-udef 43.9%

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

      3. associate-/l/ 43.9%

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

   5. Applied egg-rr 43.9%

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

      1. expm1-def 86.8%

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

      2. expm1-log1p 95.9%

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

      3. *-commutative 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in t1 around inf 82.2%

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

      1. +-commutative 82.2%

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

      2. mul-1-neg 82.2%

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

      3. unsub-neg 82.2%

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

      4. *-commutative 82.2%

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

   10. Simplified 82.2%

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

   if -9e4 < t1 < 7.1000000000000001e-62

   1. Initial program 85.5%

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

      1. distribute-lft-neg-out 85.5%

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

      2. distribute-rgt-neg-in 85.5%

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

      3. associate-*r/ 81.4%

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

      4. neg-mul-1 81.4%

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

      5. *-commutative 81.4%

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

      6. associate-/l* 81.4%

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

      7. *-rgt-identity 81.4%

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

      8. associate-*r/ 81.4%

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

      9. *-inverses 81.4%

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

      10. associate-*r* 81.4%

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

      11. times-frac 75.5%

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

      12. *-commutative 75.5%

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

      13. neg-mul-1 75.5%

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

      14. associate-/l* 80.5%

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

      15. associate-/r/ 81.4%

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

      16. neg-mul-1 81.4%

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

      17. *-commutative 81.4%

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

      18. associate-/r* 81.4%

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

      19. *-inverses 81.4%

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

      20. metadata-eval 81.4%

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

      21. neg-mul-1 81.4%

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

      22. distribute-neg-in 81.4%

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

      23. sub-neg 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in t1 around 0 77.2%

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

      1. associate-*r/ 77.2%

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

      2. neg-mul-1 77.2%

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

      3. unpow2 77.2%

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

   6. Simplified 77.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -90000 \lor \neg \left(t1 \leq 7.1 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \]

Alternative 4: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -90000 \lor \neg \left(t1 \leq 7.2 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -90000.0) (not (<= t1 7.2e-62)))
   (/ v (- (* u -2.0) t1))
   (* t1 (/ (/ v (- u)) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -90000.0) || !(t1 <= 7.2e-62)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t1 * ((v / -u) / 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 <= (-90000.0d0)) .or. (.not. (t1 <= 7.2d-62))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = t1 * ((v / -u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -90000.0) || !(t1 <= 7.2e-62)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t1 * ((v / -u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -90000.0) or not (t1 <= 7.2e-62):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = t1 * ((v / -u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -90000.0) || !(t1 <= 7.2e-62))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(-u)) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -90000.0) || ~((t1 <= 7.2e-62)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = t1 * ((v / -u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -90000.0], N[Not[LessEqual[t1, 7.2e-62]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / (-u)), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9e4 or 7.1999999999999999e-62 < t1

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 90.8%

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

      2. expm1-udef 43.9%

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

      3. associate-/l/ 43.9%

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

   5. Applied egg-rr 43.9%

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

      1. expm1-def 86.8%

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

      2. expm1-log1p 95.9%

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

      3. *-commutative 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in t1 around inf 82.2%

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

      1. +-commutative 82.2%

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

      2. mul-1-neg 82.2%

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

      3. unsub-neg 82.2%

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

      4. *-commutative 82.2%

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

   10. Simplified 82.2%

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

   if -9e4 < t1 < 7.1999999999999999e-62

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. times-frac 95.8%

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

      3. neg-mul-1 95.8%

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

      4. associate-/l* 95.8%

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

      5. associate-*r/ 95.8%

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

      6. associate-/l* 95.8%

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

      7. associate-/l/ 95.8%

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

      8. neg-mul-1 95.8%

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

      9. *-lft-identity 95.8%

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

      10. metadata-eval 95.8%

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

      11. times-frac 95.8%

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

      12. neg-mul-1 95.8%

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

      13. remove-double-neg 95.8%

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

      14. neg-mul-1 95.8%

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

      15. sub0-neg 95.8%

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

      16. associate--r+ 95.8%

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

      17. neg-sub0 95.8%

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

      18. div-sub 95.8%

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

      19. distribute-frac-neg 95.8%

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

      20. *-inverses 95.8%

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

      21. metadata-eval 95.8%

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

   3. Simplified 95.8%

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

      1. expm1-log1p-u 80.7%

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

      2. expm1-udef 57.3%

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

      3. associate-/l/ 58.0%

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

   5. Applied egg-rr 58.0%

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

      1. expm1-def 78.2%

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

      2. expm1-log1p 95.7%

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

      3. *-commutative 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in t1 around 0 78.7%

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

      1. mul-1-neg 78.7%

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

      2. unpow2 78.7%

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

   10. Simplified 78.7%

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

   11. Taylor expanded in v around 0 77.6%

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

       1. mul-1-neg 77.6%

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

       2. unpow2 77.6%

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

       3. associate-*r/ 77.2%

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

       4. distribute-rgt-neg-in 77.2%

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

       5. distribute-frac-neg 77.2%

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

       6. sqr-neg 77.2%

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

       7. associate-/r* 83.0%

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

       8. neg-mul-1 83.0%

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

       9. neg-mul-1 83.0%

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

       10. times-frac 83.0%

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

       11. metadata-eval 83.0%

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

       12. *-lft-identity 83.0%

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

       13. associate-/r* 77.2%

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

       14. associate-/l/ 83.0%

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

   13. Simplified 83.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -90000 \lor \neg \left(t1 \leq 7.2 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{u}\\ \end{array} \]

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -95000 \lor \neg \left(t1 \leq 1.5 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -95000.0) (not (<= t1 1.5e-61)))
   (/ v (- (* u -2.0) t1))
   (* (/ (- t1) u) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -95000.0) || !(t1 <= 1.5e-61)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-t1 / u) * (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 <= (-95000.0d0)) .or. (.not. (t1 <= 1.5d-61))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (-t1 / u) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -95000.0) || !(t1 <= 1.5e-61)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-t1 / u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -95000.0) or not (t1 <= 1.5e-61):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (-t1 / u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -95000.0) || !(t1 <= 1.5e-61))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -95000.0) || ~((t1 <= 1.5e-61)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (-t1 / u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -95000.0], N[Not[LessEqual[t1, 1.5e-61]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -95000 or 1.50000000000000006e-61 < t1

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 90.8%

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

      2. expm1-udef 43.9%

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

      3. associate-/l/ 43.9%

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

   5. Applied egg-rr 43.9%

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

      1. expm1-def 86.8%

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

      2. expm1-log1p 95.9%

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

      3. *-commutative 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in t1 around inf 82.2%

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

      1. +-commutative 82.2%

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

      2. mul-1-neg 82.2%

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

      3. unsub-neg 82.2%

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

      4. *-commutative 82.2%

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

   10. Simplified 82.2%

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

   if -95000 < t1 < 1.50000000000000006e-61

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. times-frac 95.8%

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

      3. neg-mul-1 95.8%

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

      4. associate-/l* 95.8%

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

      5. associate-*r/ 95.8%

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

      6. associate-/l* 95.8%

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

      7. associate-/l/ 95.8%

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

      8. neg-mul-1 95.8%

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

      9. *-lft-identity 95.8%

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

      10. metadata-eval 95.8%

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

      11. times-frac 95.8%

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

      12. neg-mul-1 95.8%

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

      13. remove-double-neg 95.8%

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

      14. neg-mul-1 95.8%

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

      15. sub0-neg 95.8%

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

      16. associate--r+ 95.8%

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

      17. neg-sub0 95.8%

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

      18. div-sub 95.8%

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

      19. distribute-frac-neg 95.8%

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

      20. *-inverses 95.8%

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

      21. metadata-eval 95.8%

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

   3. Simplified 95.8%

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

      1. clear-num 95.7%

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

      2. inv-pow 95.7%

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

   5. Applied egg-rr 95.7%

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

      1. unpow-1 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in t1 around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. times-frac 84.9%

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

      4. associate-*r* 84.9%

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

      5. associate-*r/ 84.9%

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

      6. neg-mul-1 84.9%

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

   10. Simplified 84.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -95000 \lor \neg \left(t1 \leq 1.5 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 6: 67.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.6 \cdot 10^{+129} \lor \neg \left(u \leq 2.7 \cdot 10^{+82}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7.6e+129) (not (<= u 2.7e+82)))
   (* v (/ t1 (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.6e+129) || !(u <= 2.7e+82)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-7.6d+129)) .or. (.not. (u <= 2.7d+82))) then
        tmp = v * (t1 / (u * u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.6e+129) || !(u <= 2.7e+82)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -7.6e+129) or not (u <= 2.7e+82):
		tmp = v * (t1 / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -7.6e+129) || !(u <= 2.7e+82))
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -7.6e+129) || ~((u <= 2.7e+82)))
		tmp = v * (t1 / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -7.6e+129], N[Not[LessEqual[u, 2.7e+82]], $MachinePrecision]], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -7.60000000000000011e129 or 2.6999999999999999e82 < u

   1. Initial program 80.6%

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

      1. distribute-lft-neg-out 80.6%

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

      2. distribute-rgt-neg-in 80.6%

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

      3. associate-*r/ 79.1%

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

      4. neg-mul-1 79.1%

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

      5. *-commutative 79.1%

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

      6. associate-/l* 79.1%

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

      7. *-rgt-identity 79.1%

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

      8. associate-*r/ 79.1%

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

      9. *-inverses 79.1%

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

      10. associate-*r* 79.1%

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

      11. times-frac 79.1%

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

      12. *-commutative 79.1%

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

      13. neg-mul-1 79.1%

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

      14. associate-/l* 79.1%

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

      15. associate-/r/ 79.1%

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

      16. neg-mul-1 79.1%

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

      17. *-commutative 79.1%

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

      18. associate-/r* 79.1%

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

      19. *-inverses 79.1%

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

      20. metadata-eval 79.1%

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

      21. neg-mul-1 79.1%

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

      22. distribute-neg-in 79.1%

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

      23. sub-neg 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in t1 around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. neg-mul-1 78.0%

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

      3. unpow2 78.0%

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

   6. Simplified 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*l/ 79.2%

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

      3. add-sqr-sqrt 43.0%

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

      4. sqrt-unprod 68.3%

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

      5. sqr-neg 68.3%

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

      6. sqrt-unprod 33.9%

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

      7. add-sqr-sqrt 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-/l* 73.1%

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

      3. add-sqr-sqrt 34.1%

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

      4. sqrt-unprod 65.0%

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

      5. sqr-neg 65.0%

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

      6. sqrt-unprod 42.3%

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

      7. add-sqr-sqrt 77.7%

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

      8. associate-/r/ 77.3%

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

      9. add-sqr-sqrt 41.9%

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

      10. sqrt-unprod 66.8%

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

      11. sqr-neg 66.8%

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

      12. sqrt-unprod 34.1%

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

      13. add-sqr-sqrt 73.1%

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

   10. Applied egg-rr 73.1%

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

   if -7.60000000000000011e129 < u < 2.6999999999999999e82

   1. Initial program 67.2%

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

      1. times-frac 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in t1 around inf 68.6%

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

      1. associate-*r/ 68.6%

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

      2. neg-mul-1 68.6%

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

   6. Simplified 68.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.6 \cdot 10^{+129} \lor \neg \left(u \leq 2.7 \cdot 10^{+82}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 7: 68.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+99} \lor \neg \left(u \leq 3.8 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.4e+99) (not (<= u 3.8e+85)))
   (/ t1 (/ (* u u) v))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.4e+99) || !(u <= 3.8e+85)) {
		tmp = t1 / ((u * u) / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.4d+99)) .or. (.not. (u <= 3.8d+85))) then
        tmp = t1 / ((u * u) / v)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.4e+99) || !(u <= 3.8e+85)) {
		tmp = t1 / ((u * u) / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.4e+99) or not (u <= 3.8e+85):
		tmp = t1 / ((u * u) / v)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.4e+99) || !(u <= 3.8e+85))
		tmp = Float64(t1 / Float64(Float64(u * u) / v));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.4e+99) || ~((u <= 3.8e+85)))
		tmp = t1 / ((u * u) / v);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.4e+99], N[Not[LessEqual[u, 3.8e+85]], $MachinePrecision]], N[(t1 / N[(N[(u * u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.4e99 or 3.79999999999999992e85 < u

   1. Initial program 80.1%

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

      1. distribute-lft-neg-out 80.1%

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

      2. distribute-rgt-neg-in 80.1%

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

      3. associate-*r/ 79.8%

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

      4. neg-mul-1 79.8%

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

      5. *-commutative 79.8%

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

      6. associate-/l* 79.8%

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

      7. *-rgt-identity 79.8%

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

      8. associate-*r/ 79.8%

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

      9. *-inverses 79.8%

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

      10. associate-*r* 79.8%

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

      11. times-frac 79.8%

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

      12. *-commutative 79.8%

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

      13. neg-mul-1 79.8%

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

      14. associate-/l* 78.8%

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

      15. associate-/r/ 79.8%

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

      16. neg-mul-1 79.8%

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

      17. *-commutative 79.8%

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

      18. associate-/r* 79.8%

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

      19. *-inverses 79.8%

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

      20. metadata-eval 79.8%

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

      21. neg-mul-1 79.8%

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

      22. distribute-neg-in 79.8%

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

      23. sub-neg 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in t1 around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. neg-mul-1 77.1%

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

      3. unpow2 77.1%

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

   6. Simplified 77.1%

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

      1. clear-num 77.7%

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

      2. un-div-inv 77.7%

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

      3. add-sqr-sqrt 42.8%

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

      4. sqrt-unprod 67.5%

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

      5. sqr-neg 67.5%

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

      6. sqrt-unprod 32.4%

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

      7. add-sqr-sqrt 71.2%

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

   8. Applied egg-rr 71.2%

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

   if -1.4e99 < u < 3.79999999999999992e85

   1. Initial program 66.5%

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

      1. times-frac 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in t1 around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. neg-mul-1 69.9%

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

   6. Simplified 69.9%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+99} \lor \neg \left(u \leq 3.8 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return (v / (t1 + u)) / (-1.0 - (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 / (t1 + u)) / ((-1.0d0) - (u / t1))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. *-commutative 71.6%

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

   2. times-frac 98.1%

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

   3. neg-mul-1 98.1%

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

   4. associate-/l* 98.1%

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

   5. associate-*r/ 98.1%

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

   6. associate-/l* 98.1%

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

   7. associate-/l/ 98.1%

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

   8. neg-mul-1 98.1%

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

   9. *-lft-identity 98.1%

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

   10. metadata-eval 98.1%

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

   11. times-frac 98.1%

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

   12. neg-mul-1 98.1%

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

   13. remove-double-neg 98.1%

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

   14. neg-mul-1 98.1%

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

   15. sub0-neg 98.1%

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

   16. associate--r+ 98.1%

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

   17. neg-sub0 98.1%

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

   18. div-sub 98.1%

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

   19. distribute-frac-neg 98.1%

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

   20. *-inverses 98.1%

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

   21. metadata-eval 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 9: 57.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.5 \cdot 10^{+242} \lor \neg \left(u \leq 4.7 \cdot 10^{+85}\right):\\ \;\;\;\;-0.5 \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.5e+242) (not (<= u 4.7e+85))) (* -0.5 (/ v u)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.5e+242) || !(u <= 4.7e+85)) {
		tmp = -0.5 * (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 <= (-4.5d+242)) .or. (.not. (u <= 4.7d+85))) then
        tmp = (-0.5d0) * (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 <= -4.5e+242) || !(u <= 4.7e+85)) {
		tmp = -0.5 * (v / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.5e+242) or not (u <= 4.7e+85):
		tmp = -0.5 * (v / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.5e+242) || !(u <= 4.7e+85))
		tmp = Float64(-0.5 * 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 <= -4.5e+242) || ~((u <= 4.7e+85)))
		tmp = -0.5 * (v / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.5e+242], N[Not[LessEqual[u, 4.7e+85]], $MachinePrecision]], N[(-0.5 * N[(v / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.4999999999999996e242 or 4.7000000000000002e85 < u

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

      2. times-frac 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 99.9%

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

      2. expm1-udef 76.8%

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

      3. associate-/l/ 76.8%

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

   5. Applied egg-rr 76.8%

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

      1. expm1-def 89.2%

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

      2. expm1-log1p 89.2%

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

      3. *-commutative 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in t1 around inf 39.3%

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

      1. +-commutative 39.3%

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

      2. mul-1-neg 39.3%

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

      3. unsub-neg 39.3%

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

      4. *-commutative 39.3%

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

   10. Simplified 39.3%

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

   11. Taylor expanded in u around inf 35.0%

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

   if -4.4999999999999996e242 < u < 4.7000000000000002e85

   1. Initial program 67.3%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. neg-mul-1 64.2%

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

   6. Simplified 64.2%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.5 \cdot 10^{+242} \lor \neg \left(u \leq 4.7 \cdot 10^{+85}\right):\\ \;\;\;\;-0.5 \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10: 57.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{+243} \lor \neg \left(u \leq 4.7 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.9e+243) (not (<= u 4.7e+85))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.9e+243) || !(u <= 4.7e+85)) {
		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 <= (-2.9d+243)) .or. (.not. (u <= 4.7d+85))) 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 <= -2.9e+243) || !(u <= 4.7e+85)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.9e+243) or not (u <= 4.7e+85):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.9e+243) || !(u <= 4.7e+85))
		tmp = Float64(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 <= -2.9e+243) || ~((u <= 4.7e+85)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.9e+243], N[Not[LessEqual[u, 4.7e+85]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.90000000000000006e243 or 4.7000000000000002e85 < u

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

      2. times-frac 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 94.5%

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

   5. Taylor expanded in u around 0 35.0%

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

      1. associate-*r/ 35.0%

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

      2. neg-mul-1 35.0%

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

   7. Simplified 35.0%

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

   if -2.90000000000000006e243 < u < 4.7000000000000002e85

   1. Initial program 67.3%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. neg-mul-1 64.2%

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

   6. Simplified 64.2%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{+243} \lor \neg \left(u \leq 4.7 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 61.8% 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 71.6%

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

   1. times-frac 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in t1 around inf 59.5%

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

5. Final simplification 59.5%

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

Alternative 12: 54.0% accurate, 3.0× 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(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 71.6%

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

   1. times-frac 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in t1 around inf 51.6%

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

   1. associate-*r/ 51.6%

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

   2. neg-mul-1 51.6%

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

6. Simplified 51.6%

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

7. Final simplification 51.6%

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

Reproduce

herbie shell --seed 2023171 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))