Rosa's DopplerBench

Percentage Accurate: 72.8% → 98.4%

Time: 10.1s

Alternatives: 14

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: 72.8% 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: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. times-frac 97.7%

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

   2. neg-mul-1 97.7%

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

   3. associate-/l* 97.6%

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

   4. associate-*l/ 97.7%

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

   5. neg-mul-1 97.7%

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

   6. distribute-frac-neg 97.7%

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

   7. +-commutative 97.7%

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

   8. remove-double-neg 97.7%

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

   9. unsub-neg 97.7%

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

   10. div-sub 97.7%

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

   11. sub-neg 97.7%

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

   12. distribute-frac-neg 97.7%

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

   13. remove-double-neg 97.7%

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

   14. *-inverses 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in v around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. +-commutative 96.5%

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

   3. *-commutative 96.5%

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

   4. distribute-neg-frac 96.5%

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

6. Simplified 96.5%

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

   1. neg-mul-1 96.5%

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

   2. times-frac 97.8%

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

8. Applied egg-rr 97.8%

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

   1. associate-*l/ 98.0%

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

   2. associate-*r/ 98.0%

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

   3. neg-mul-1 98.0%

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

   4. +-commutative 98.0%

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

10. Applied egg-rr 98.0%

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

    1. frac-2neg 98.0%

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

    2. div-inv 97.9%

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

    3. remove-double-neg 97.9%

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

    4. +-commutative 97.9%

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

    5. distribute-neg-in 97.9%

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

    6. metadata-eval 97.9%

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

12. Applied egg-rr 97.9%

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

    1. associate-*r/ 98.0%

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

    2. *-rgt-identity 98.0%

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

    3. unsub-neg 98.0%

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

14. Simplified 98.0%

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

15. Final simplification 98.0%

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

Alternative 2: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ t_2 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;t1 \leq -1.05 \cdot 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 3.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{-v}{t1} \cdot \frac{t1}{u + t1}\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (/ t1 u) (- t1 u)))) (t_2 (/ (- v) (+ t1 (* u 2.0)))))
   (if (<= t1 -1.05e-38)
     t_2
     (if (<= t1 2.6e-130)
       t_1
       (if (<= t1 3.5e-91)
         (* (/ (- v) t1) (/ t1 (+ u t1)))
         (if (<= t1 4.5e-33) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * ((t1 / u) / (t1 - u));
	double t_2 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -1.05e-38) {
		tmp = t_2;
	} else if (t1 <= 2.6e-130) {
		tmp = t_1;
	} else if (t1 <= 3.5e-91) {
		tmp = (-v / t1) * (t1 / (u + t1));
	} else if (t1 <= 4.5e-33) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v * ((t1 / u) / (t1 - u))
    t_2 = -v / (t1 + (u * 2.0d0))
    if (t1 <= (-1.05d-38)) then
        tmp = t_2
    else if (t1 <= 2.6d-130) then
        tmp = t_1
    else if (t1 <= 3.5d-91) then
        tmp = (-v / t1) * (t1 / (u + t1))
    else if (t1 <= 4.5d-33) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * ((t1 / u) / (t1 - u));
	double t_2 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -1.05e-38) {
		tmp = t_2;
	} else if (t1 <= 2.6e-130) {
		tmp = t_1;
	} else if (t1 <= 3.5e-91) {
		tmp = (-v / t1) * (t1 / (u + t1));
	} else if (t1 <= 4.5e-33) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * ((t1 / u) / (t1 - u))
	t_2 = -v / (t1 + (u * 2.0))
	tmp = 0
	if t1 <= -1.05e-38:
		tmp = t_2
	elif t1 <= 2.6e-130:
		tmp = t_1
	elif t1 <= 3.5e-91:
		tmp = (-v / t1) * (t1 / (u + t1))
	elif t1 <= 4.5e-33:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(t1 / u) / Float64(t1 - u)))
	t_2 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	tmp = 0.0
	if (t1 <= -1.05e-38)
		tmp = t_2;
	elseif (t1 <= 2.6e-130)
		tmp = t_1;
	elseif (t1 <= 3.5e-91)
		tmp = Float64(Float64(Float64(-v) / t1) * Float64(t1 / Float64(u + t1)));
	elseif (t1 <= 4.5e-33)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * ((t1 / u) / (t1 - u));
	t_2 = -v / (t1 + (u * 2.0));
	tmp = 0.0;
	if (t1 <= -1.05e-38)
		tmp = t_2;
	elseif (t1 <= 2.6e-130)
		tmp = t_1;
	elseif (t1 <= 3.5e-91)
		tmp = (-v / t1) * (t1 / (u + t1));
	elseif (t1 <= 4.5e-33)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(N[(t1 / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.05e-38], t$95$2, If[LessEqual[t1, 2.6e-130], t$95$1, If[LessEqual[t1, 3.5e-91], N[(N[((-v) / t1), $MachinePrecision] * N[(t1 / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.5e-33], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.05000000000000006e-38 or 4.49999999999999991e-33 < t1

   1. Initial program 68.7%

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

      1. times-frac 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-*l/ 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. remove-double-neg 99.9%

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

      9. unsub-neg 99.9%

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

      10. div-sub 99.9%

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

      11. sub-neg 99.9%

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

      12. distribute-frac-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 96.9%

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

      1. mul-1-neg 96.9%

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

      2. +-commutative 96.9%

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

      3. *-commutative 96.9%

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

      4. distribute-neg-frac 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in t1 around inf 79.6%

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

      1. *-commutative 79.6%

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

   9. Simplified 79.6%

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

   if -1.05000000000000006e-38 < t1 < 2.6000000000000001e-130 or 3.4999999999999999e-91 < t1 < 4.49999999999999991e-33

   1. Initial program 81.7%

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

      1. times-frac 95.0%

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

   3. Simplified 95.0%

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

      1. clear-num 95.0%

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

      2. frac-2neg 95.0%

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

      3. frac-times 95.8%

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

      4. *-un-lft-identity 95.8%

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

      5. add-sqr-sqrt 47.7%

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

      6. sqrt-unprod 50.2%

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

      7. sqr-neg 50.2%

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

      8. sqrt-unprod 18.6%

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

      9. add-sqr-sqrt 33.1%

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

      10. add-sqr-sqrt 15.4%

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

      11. sqrt-unprod 45.4%

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

      12. sqr-neg 45.4%

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

      13. sqrt-unprod 43.6%

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

      14. add-sqr-sqrt 95.8%

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

      15. distribute-neg-in 95.8%

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

      16. add-sqr-sqrt 52.1%

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

      17. sqrt-unprod 85.8%

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

      18. sqr-neg 85.8%

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

      19. sqrt-unprod 35.6%

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

      20. add-sqr-sqrt 80.6%

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

      21. sub-neg 80.6%

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

   5. Applied egg-rr 80.6%

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

   6. Taylor expanded in t1 around 0 83.0%

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

      1. clear-num 83.0%

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

      2. associate-/r/ 82.9%

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

      3. *-un-lft-identity 82.9%

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

      4. associate-/r* 83.3%

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

      5. clear-num 83.3%

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

      6. *-un-lft-identity 83.3%

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

   8. Applied egg-rr 83.3%

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

   if 2.6000000000000001e-130 < t1 < 3.4999999999999999e-91

   1. Initial program 80.7%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 74.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{-130}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;t1 \leq 3.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{-v}{t1} \cdot \frac{t1}{u + t1}\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]

Alternative 3: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u + t1}\\ \mathbf{if}\;u \leq -2.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{-t1}{u} \cdot t_1\\ \mathbf{elif}\;u \leq 1.26 \cdot 10^{-141}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot t_1}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ u t1))))
   (if (<= u -2.6e-60)
     (* (/ (- t1) u) t_1)
     (if (<= u 1.26e-141) (/ (- v) t1) (/ (* t1 t_1) (- t1 u))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (u + t1);
	double tmp;
	if (u <= -2.6e-60) {
		tmp = (-t1 / u) * t_1;
	} else if (u <= 1.26e-141) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * t_1) / (t1 - u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (u + t1)
    if (u <= (-2.6d-60)) then
        tmp = (-t1 / u) * t_1
    else if (u <= 1.26d-141) then
        tmp = -v / t1
    else
        tmp = (t1 * t_1) / (t1 - u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (u + t1);
	double tmp;
	if (u <= -2.6e-60) {
		tmp = (-t1 / u) * t_1;
	} else if (u <= 1.26e-141) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * t_1) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (u + t1)
	tmp = 0
	if u <= -2.6e-60:
		tmp = (-t1 / u) * t_1
	elif u <= 1.26e-141:
		tmp = -v / t1
	else:
		tmp = (t1 * t_1) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(u + t1))
	tmp = 0.0
	if (u <= -2.6e-60)
		tmp = Float64(Float64(Float64(-t1) / u) * t_1);
	elseif (u <= 1.26e-141)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 * t_1) / Float64(t1 - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (u + t1);
	tmp = 0.0;
	if (u <= -2.6e-60)
		tmp = (-t1 / u) * t_1;
	elseif (u <= 1.26e-141)
		tmp = -v / t1;
	else
		tmp = (t1 * t_1) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.6e-60], N[(N[((-t1) / u), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[u, 1.26e-141], N[((-v) / t1), $MachinePrecision], N[(N[(t1 * t$95$1), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.5999999999999998e-60

   1. Initial program 78.8%

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

      1. times-frac 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in t1 around 0 79.9%

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

      1. mul-1-neg 30.5%

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

      2. distribute-neg-frac 30.5%

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

   6. Simplified 79.9%

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

   if -2.5999999999999998e-60 < u < 1.26e-141

   1. Initial program 65.0%

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

      1. times-frac 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in t1 around inf 80.1%

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

      1. associate-*r/ 80.1%

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

      2. neg-mul-1 80.1%

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

   6. Simplified 80.1%

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

   if 1.26e-141 < u

   1. Initial program 81.7%

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

      1. times-frac 98.8%

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

   3. Simplified 98.8%

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

      1. frac-2neg 98.8%

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

      2. remove-double-neg 98.8%

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

      3. associate-*l/ 98.8%

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

      4. distribute-neg-in 98.8%

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

      5. add-sqr-sqrt 37.0%

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

      6. sqrt-unprod 80.1%

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

      7. sqr-neg 80.1%

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

      8. sqrt-unprod 48.3%

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

      9. add-sqr-sqrt 78.1%

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

      10. sub-neg 78.1%

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

   5. Applied egg-rr 78.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u + t1}\\ \mathbf{elif}\;u \leq 1.26 \cdot 10^{-141}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u + t1}}{t1 - u}\\ \end{array} \]

Alternative 4: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-38} \lor \neg \left(t1 \leq 6.8 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.2e-38) (not (<= t1 6.8e-32)))
   (/ (- v) (+ t1 (* u 2.0)))
   (* (/ t1 u) (/ v (- t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.2e-38) || !(t1 <= 6.8e-32)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (t1 / u) * (v / (t1 - u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.2d-38)) .or. (.not. (t1 <= 6.8d-32))) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = (t1 / u) * (v / (t1 - u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.2e-38) || !(t1 <= 6.8e-32)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (t1 / u) * (v / (t1 - u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.2e-38) or not (t1 <= 6.8e-32):
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = (t1 / u) * (v / (t1 - u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.2e-38) || !(t1 <= 6.8e-32))
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(Float64(t1 / u) * Float64(v / Float64(t1 - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.2e-38) || ~((t1 <= 6.8e-32)))
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = (t1 / u) * (v / (t1 - u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.2e-38], N[Not[LessEqual[t1, 6.8e-32]], $MachinePrecision]], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.20000000000000011e-38 or 6.79999999999999956e-32 < t1

   1. Initial program 68.7%

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

      1. times-frac 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-*l/ 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. remove-double-neg 99.9%

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

      9. unsub-neg 99.9%

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

      10. div-sub 99.9%

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

      11. sub-neg 99.9%

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

      12. distribute-frac-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 96.9%

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

      1. mul-1-neg 96.9%

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

      2. +-commutative 96.9%

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

      3. *-commutative 96.9%

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

      4. distribute-neg-frac 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in t1 around inf 79.6%

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

      1. *-commutative 79.6%

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

   9. Simplified 79.6%

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

   if -1.20000000000000011e-38 < t1 < 6.79999999999999956e-32

   1. Initial program 81.6%

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

      1. times-frac 95.6%

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

   3. Simplified 95.6%

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

      1. clear-num 95.5%

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

      2. frac-2neg 95.5%

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

      3. frac-times 96.1%

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

      4. *-un-lft-identity 96.1%

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

      5. add-sqr-sqrt 45.3%

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

      6. sqrt-unprod 48.2%

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

      7. sqr-neg 48.2%

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

      8. sqrt-unprod 17.9%

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

      9. add-sqr-sqrt 31.6%

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

      10. add-sqr-sqrt 13.7%

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

      11. sqrt-unprod 51.4%

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

      12. sqr-neg 51.4%

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

      13. sqrt-unprod 49.7%

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

      14. add-sqr-sqrt 96.1%

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

      15. distribute-neg-in 96.1%

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

      16. add-sqr-sqrt 46.2%

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

      17. sqrt-unprod 80.5%

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

      18. sqr-neg 80.5%

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

      19. sqrt-unprod 36.0%

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

      20. add-sqr-sqrt 75.9%

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

      21. sub-neg 75.9%

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

   5. Applied egg-rr 75.9%

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

   6. Taylor expanded in t1 around 0 78.8%

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

      1. *-un-lft-identity 78.8%

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

      2. *-un-lft-identity 78.8%

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

      3. times-frac 78.3%

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

      4. clear-num 78.3%

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

      5. *-un-lft-identity 78.3%

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

   8. Applied egg-rr 78.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-38} \lor \neg \left(t1 \leq 6.8 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{t1 - u}\\ \end{array} \]

Alternative 5: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{-39} \lor \neg \left(t1 \leq 1.75 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.45e-39) (not (<= t1 1.75e-33)))
   (/ (- v) (+ t1 (* u 2.0)))
   (* v (/ (/ t1 u) (- t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.45e-39) || !(t1 <= 1.75e-33)) {
		tmp = -v / (t1 + (u * 2.0));
	} 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 ((t1 <= (-1.45d-39)) .or. (.not. (t1 <= 1.75d-33))) then
        tmp = -v / (t1 + (u * 2.0d0))
    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 ((t1 <= -1.45e-39) || !(t1 <= 1.75e-33)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = v * ((t1 / u) / (t1 - u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.45e-39) or not (t1 <= 1.75e-33):
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = v * ((t1 / u) / (t1 - u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.45e-39) || !(t1 <= 1.75e-33))
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(v * Float64(Float64(t1 / u) / Float64(t1 - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.45e-39) || ~((t1 <= 1.75e-33)))
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = v * ((t1 / u) / (t1 - u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.45e-39], N[Not[LessEqual[t1, 1.75e-33]], $MachinePrecision]], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[(t1 / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.44999999999999994e-39 or 1.7499999999999999e-33 < t1

   1. Initial program 68.7%

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

      1. times-frac 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-*l/ 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. remove-double-neg 99.9%

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

      9. unsub-neg 99.9%

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

      10. div-sub 99.9%

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

      11. sub-neg 99.9%

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

      12. distribute-frac-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 96.9%

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

      1. mul-1-neg 96.9%

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

      2. +-commutative 96.9%

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

      3. *-commutative 96.9%

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

      4. distribute-neg-frac 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in t1 around inf 79.6%

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

      1. *-commutative 79.6%

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

   9. Simplified 79.6%

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

   if -1.44999999999999994e-39 < t1 < 1.7499999999999999e-33

   1. Initial program 81.6%

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

      1. times-frac 95.6%

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

   3. Simplified 95.6%

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

      1. clear-num 95.5%

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

      2. frac-2neg 95.5%

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

      3. frac-times 96.1%

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

      4. *-un-lft-identity 96.1%

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

      5. add-sqr-sqrt 45.3%

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

      6. sqrt-unprod 48.2%

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

      7. sqr-neg 48.2%

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

      8. sqrt-unprod 17.9%

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

      9. add-sqr-sqrt 31.6%

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

      10. add-sqr-sqrt 13.7%

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

      11. sqrt-unprod 51.4%

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

      12. sqr-neg 51.4%

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

      13. sqrt-unprod 49.7%

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

      14. add-sqr-sqrt 96.1%

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

      15. distribute-neg-in 96.1%

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

      16. add-sqr-sqrt 46.2%

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

      17. sqrt-unprod 80.5%

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

      18. sqr-neg 80.5%

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

      19. sqrt-unprod 36.0%

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

      20. add-sqr-sqrt 75.9%

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

      21. sub-neg 75.9%

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

   5. Applied egg-rr 75.9%

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

   6. Taylor expanded in t1 around 0 78.8%

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

      1. clear-num 78.8%

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

      2. associate-/r/ 78.7%

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

      3. *-un-lft-identity 78.7%

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

      4. associate-/r* 79.1%

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

      5. clear-num 79.1%

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

      6. *-un-lft-identity 79.1%

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

   8. Applied egg-rr 79.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{-39} \lor \neg \left(t1 \leq 1.75 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \end{array} \]

Alternative 6: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. times-frac 97.7%

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

   2. neg-mul-1 97.7%

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

   3. associate-/l* 97.6%

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

   4. associate-*l/ 97.7%

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

   5. neg-mul-1 97.7%

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

   6. distribute-frac-neg 97.7%

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

   7. +-commutative 97.7%

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

   8. remove-double-neg 97.7%

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

   9. unsub-neg 97.7%

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

   10. div-sub 97.7%

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

   11. sub-neg 97.7%

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

   12. distribute-frac-neg 97.7%

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

   13. remove-double-neg 97.7%

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

   14. *-inverses 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in v around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. +-commutative 96.5%

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

   3. *-commutative 96.5%

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

   4. distribute-neg-frac 96.5%

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

6. Simplified 96.5%

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

   1. neg-mul-1 96.5%

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

   2. times-frac 97.8%

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

8. Applied egg-rr 97.8%

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

9. Taylor expanded in v around 0 96.5%

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

    1. *-commutative 96.5%

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

    2. associate-/r* 98.0%

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

    3. +-commutative 98.0%

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

    4. +-commutative 98.0%

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

    5. metadata-eval 98.0%

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

    6. times-frac 98.0%

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

    7. *-rgt-identity 98.0%

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

    8. associate-/l/ 96.5%

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

    9. associate-*l* 96.5%

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

    10. neg-mul-1 96.5%

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

    11. +-commutative 96.5%

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

    12. distribute-neg-in 96.5%

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

    13. metadata-eval 96.5%

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

    14. associate-/r* 97.7%

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

    15. unsub-neg 97.7%

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

11. Simplified 97.7%

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

12. Final simplification 97.7%

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

Alternative 7: 59.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+144}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.6e+246)
   (/ (- v) u)
   (if (<= u 9e+144) (/ (- v) t1) (/ 1.0 (/ (- u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.6e+246) {
		tmp = -v / u;
	} else if (u <= 9e+144) {
		tmp = -v / t1;
	} else {
		tmp = 1.0 / (-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 <= (-5.6d+246)) then
        tmp = -v / u
    else if (u <= 9d+144) then
        tmp = -v / t1
    else
        tmp = 1.0d0 / (-u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.6e+246) {
		tmp = -v / u;
	} else if (u <= 9e+144) {
		tmp = -v / t1;
	} else {
		tmp = 1.0 / (-u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.6e+246:
		tmp = -v / u
	elif u <= 9e+144:
		tmp = -v / t1
	else:
		tmp = 1.0 / (-u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.6e+246)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 9e+144)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(1.0 / Float64(Float64(-u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.6e+246)
		tmp = -v / u;
	elseif (u <= 9e+144)
		tmp = -v / t1;
	else
		tmp = 1.0 / (-u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.6e+246], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 9e+144], N[((-v) / t1), $MachinePrecision], N[(1.0 / N[((-u) / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.59999999999999976e246

   1. Initial program 94.6%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 57.8%

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

   5. Taylor expanded in t1 around 0 42.5%

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

      1. associate-*r/ 42.5%

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

      2. neg-mul-1 42.5%

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

   7. Simplified 42.5%

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

   if -5.59999999999999976e246 < u < 8.99999999999999935e144

   1. Initial program 72.4%

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

      1. times-frac 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t1 around inf 55.7%

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

      1. associate-*r/ 55.7%

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

      2. neg-mul-1 55.7%

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

   6. Simplified 55.7%

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

   if 8.99999999999999935e144 < u

   1. Initial program 84.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 90.8%

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

      4. *-un-lft-identity 90.8%

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

      5. add-sqr-sqrt 49.9%

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

      6. sqrt-unprod 84.3%

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

      7. sqr-neg 84.3%

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

      8. sqrt-unprod 37.7%

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

      9. add-sqr-sqrt 81.3%

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

      10. add-sqr-sqrt 23.3%

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

      11. sqrt-unprod 77.3%

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

      12. sqr-neg 77.3%

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

      13. sqrt-unprod 67.5%

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

      14. add-sqr-sqrt 90.8%

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

      15. distribute-neg-in 90.8%

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

      16. add-sqr-sqrt 23.3%

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

      17. sqrt-unprod 87.7%

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

      18. sqr-neg 87.7%

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

      19. sqrt-unprod 67.5%

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

      20. add-sqr-sqrt 90.8%

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

      21. sub-neg 90.8%

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

   5. Applied egg-rr 90.8%

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

   6. Taylor expanded in t1 around inf 52.8%

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

      1. clear-num 53.4%

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

      2. inv-pow 53.4%

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

      3. *-un-lft-identity 53.4%

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

   8. Applied egg-rr 53.4%

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

      1. unpow-1 53.4%

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

   10. Simplified 53.4%

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

   11. Taylor expanded in t1 around 0 53.4%

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

       1. neg-mul-1 53.4%

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

       2. distribute-neg-frac 53.4%

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

   13. Simplified 53.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{-u}{v}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+144}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-u}{v}}\\ \end{array} \]

Alternative 8: 63.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 5.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 5.2e+147) (/ (- v) (+ u t1)) (* (/ (- t1) u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 5.2e+147) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 / u) * (v / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 5.2d+147) then
        tmp = -v / (u + t1)
    else
        tmp = (-t1 / u) * (v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 5.2e+147) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 / u) * (v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 5.2e+147:
		tmp = -v / (u + t1)
	else:
		tmp = (-t1 / u) * (v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 5.2e+147)
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 5.2e+147)
		tmp = -v / (u + t1);
	else
		tmp = (-t1 / u) * (v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 5.2e+147], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 5.1999999999999997e147

   1. Initial program 74.4%

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

      1. times-frac 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in t1 around inf 58.6%

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

   5. Taylor expanded in v around 0 55.4%

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

      1. associate-*r/ 55.4%

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

      2. neg-mul-1 55.4%

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

      3. +-commutative 55.4%

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

   7. Simplified 55.4%

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

   if 5.1999999999999997e147 < u

   1. Initial program 83.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around inf 73.0%

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

   5. Taylor expanded in t1 around 0 73.0%

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

      1. mul-1-neg 73.0%

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

      2. distribute-neg-frac 73.0%

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

   7. Simplified 73.0%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 5.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{t1}\\ \end{array} \]

Alternative 9: 63.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 10^{+154}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 1e+154) (/ (- v) (+ t1 (* u 2.0))) (* (/ (- t1) u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1e+154) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (-t1 / u) * (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 <= 1d+154) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = (-t1 / u) * (v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1e+154) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (-t1 / u) * (v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 1e+154:
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = (-t1 / u) * (v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 1e+154)
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 1e+154)
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = (-t1 / u) * (v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 1e+154], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 1.00000000000000004e154

   1. Initial program 74.5%

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

      1. times-frac 97.4%

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

      2. neg-mul-1 97.4%

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

      3. associate-/l* 97.4%

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

      4. associate-*l/ 97.4%

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

      5. neg-mul-1 97.4%

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

      6. distribute-frac-neg 97.4%

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

      7. +-commutative 97.4%

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

      8. remove-double-neg 97.4%

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

      9. unsub-neg 97.4%

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

      10. div-sub 97.4%

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

      11. sub-neg 97.4%

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

      12. distribute-frac-neg 97.4%

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

      13. remove-double-neg 97.4%

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

      14. *-inverses 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in v around 0 97.2%

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

      1. mul-1-neg 97.2%

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

      2. +-commutative 97.2%

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

      3. *-commutative 97.2%

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

      4. distribute-neg-frac 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in t1 around inf 56.1%

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

      1. *-commutative 56.1%

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

   9. Simplified 56.1%

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

   if 1.00000000000000004e154 < u

   1. Initial program 82.4%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 75.5%

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

   5. Taylor expanded in t1 around 0 75.5%

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

      1. mul-1-neg 75.5%

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

      2. distribute-neg-frac 75.5%

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

   7. Simplified 75.5%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 10^{+154}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{t1}\\ \end{array} \]

Alternative 10: 58.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+246} \lor \neg \left(u \leq 6.6 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.6e+246) (not (<= u 6.6e+146))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.6e+246) || !(u <= 6.6e+146)) {
		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 <= (-5.6d+246)) .or. (.not. (u <= 6.6d+146))) 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 <= -5.6e+246) || !(u <= 6.6e+146)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.6e+246) or not (u <= 6.6e+146):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.6e+246) || !(u <= 6.6e+146))
		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 <= -5.6e+246) || ~((u <= 6.6e+146)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.6e+246], N[Not[LessEqual[u, 6.6e+146]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.59999999999999976e246 or 6.60000000000000032e146 < u

   1. Initial program 87.9%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 92.1%

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

      4. *-un-lft-identity 92.1%

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

      5. add-sqr-sqrt 46.8%

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

      6. sqrt-unprod 85.8%

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

      7. sqr-neg 85.8%

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

      8. sqrt-unprod 43.2%

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

      9. add-sqr-sqrt 88.0%

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

      10. add-sqr-sqrt 36.4%

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

      11. sqrt-unprod 79.3%

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

      12. sqr-neg 79.3%

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

      13. sqrt-unprod 55.7%

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

      14. add-sqr-sqrt 92.1%

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

      15. distribute-neg-in 92.1%

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

      16. add-sqr-sqrt 36.4%

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

      17. sqrt-unprod 90.1%

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

      18. sqr-neg 90.1%

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

      19. sqrt-unprod 55.7%

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

      20. add-sqr-sqrt 92.1%

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

      21. sub-neg 92.1%

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

   5. Applied egg-rr 92.1%

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

   6. Taylor expanded in t1 around 0 92.1%

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

   7. Taylor expanded in u around 0 49.6%

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

   if -5.59999999999999976e246 < u < 6.60000000000000032e146

   1. Initial program 72.5%

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

      1. times-frac 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in t1 around inf 55.5%

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

      1. associate-*r/ 55.5%

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

      2. neg-mul-1 55.5%

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

   6. Simplified 55.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+246} \lor \neg \left(u \leq 6.6 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 58.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+248} \lor \neg \left(u \leq 8 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.2e+248) (not (<= u 8e+144))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.2e+248) || !(u <= 8e+144)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.2d+248)) .or. (.not. (u <= 8d+144))) then
        tmp = -v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.2e+248) || !(u <= 8e+144)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.2e+248) or not (u <= 8e+144):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.2e+248) || !(u <= 8e+144))
		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 <= -3.2e+248) || ~((u <= 8e+144)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.2e+248], N[Not[LessEqual[u, 8e+144]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.19999999999999985e248 or 8.00000000000000019e144 < u

   1. Initial program 88.1%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 64.4%

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

   5. Taylor expanded in t1 around 0 48.9%

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

      1. associate-*r/ 48.9%

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

      2. neg-mul-1 48.9%

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

   7. Simplified 48.9%

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

   if -3.19999999999999985e248 < u < 8.00000000000000019e144

   1. Initial program 72.4%

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

      1. times-frac 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t1 around inf 55.7%

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

      1. associate-*r/ 55.7%

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

      2. neg-mul-1 55.7%

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

   6. Simplified 55.7%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+248} \lor \neg \left(u \leq 8 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12: 24.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6.8 \cdot 10^{+66} \lor \neg \left(t1 \leq 1.4 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -6.8e+66) (not (<= t1 1.4e+111))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6.8e+66) || !(t1 <= 1.4e+111)) {
		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 <= (-6.8d+66)) .or. (.not. (t1 <= 1.4d+111))) 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 <= -6.8e+66) || !(t1 <= 1.4e+111)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -6.8e+66) or not (t1 <= 1.4e+111):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -6.8e+66) || !(t1 <= 1.4e+111))
		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 <= -6.8e+66) || ~((t1 <= 1.4e+111)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -6.8e+66], N[Not[LessEqual[t1, 1.4e+111]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.8000000000000006e66 or 1.4e111 < t1

   1. Initial program 56.9%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 96.5%

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

      4. *-un-lft-identity 96.5%

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

      5. add-sqr-sqrt 37.0%

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

      6. sqrt-unprod 55.1%

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

      7. sqr-neg 55.1%

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

      8. sqrt-unprod 26.4%

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

      9. add-sqr-sqrt 44.3%

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

      10. add-sqr-sqrt 19.0%

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

      11. sqrt-unprod 23.6%

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

      12. sqr-neg 23.6%

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

      13. sqrt-unprod 58.0%

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

      14. add-sqr-sqrt 96.5%

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

      15. distribute-neg-in 96.5%

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

      16. add-sqr-sqrt 38.1%

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

      17. sqrt-unprod 50.9%

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

      18. sqr-neg 50.9%

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

      19. sqrt-unprod 29.1%

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

      20. add-sqr-sqrt 50.3%

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

      21. sub-neg 50.3%

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

   5. Applied egg-rr 50.3%

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

   6. Taylor expanded in t1 around inf 40.6%

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

   if -6.8000000000000006e66 < t1 < 1.4e111

   1. Initial program 83.9%

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

      1. times-frac 96.6%

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

   3. Simplified 96.6%

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

      1. clear-num 96.6%

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

      2. frac-2neg 96.6%

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

      3. frac-times 96.4%

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

      4. *-un-lft-identity 96.4%

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

      5. add-sqr-sqrt 43.8%

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

      6. sqrt-unprod 47.0%

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

      7. sqr-neg 47.0%

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

      8. sqrt-unprod 17.7%

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

      9. add-sqr-sqrt 32.1%

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

      10. add-sqr-sqrt 13.1%

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

      11. sqrt-unprod 55.8%

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

      12. sqr-neg 55.8%

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

      13. sqrt-unprod 51.7%

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

      14. add-sqr-sqrt 96.4%

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

      15. distribute-neg-in 96.4%

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

      16. add-sqr-sqrt 44.5%

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

      17. sqrt-unprod 78.2%

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

      18. sqr-neg 78.2%

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

      19. sqrt-unprod 34.9%

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

      20. add-sqr-sqrt 69.4%

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

      21. sub-neg 69.4%

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

   5. Applied egg-rr 69.4%

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

   6. Taylor expanded in t1 around 0 71.2%

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

   7. Taylor expanded in u around 0 17.6%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6.8 \cdot 10^{+66} \lor \neg \left(t1 \leq 1.4 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 13: 62.9% accurate, 2.0× 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(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 75.3%

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

   1. times-frac 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in t1 around inf 60.2%

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

5. Taylor expanded in v around 0 55.5%

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

   1. associate-*r/ 55.5%

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

   2. neg-mul-1 55.5%

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

   3. +-commutative 55.5%

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

7. Simplified 55.5%

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

8. Final simplification 55.5%

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

Alternative 14: 14.5% 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 75.3%

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

   1. times-frac 97.7%

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

3. Simplified 97.7%

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

   1. clear-num 97.6%

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

   2. frac-2neg 97.6%

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

   3. frac-times 96.5%

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

   4. *-un-lft-identity 96.5%

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

   5. add-sqr-sqrt 41.7%

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

   6. sqrt-unprod 49.6%

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

   7. sqr-neg 49.6%

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

   8. sqrt-unprod 20.5%

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

   9. add-sqr-sqrt 36.0%

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

   10. add-sqr-sqrt 15.0%

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

   11. sqrt-unprod 45.6%

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

   12. sqr-neg 45.6%

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

   13. sqrt-unprod 53.7%

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

   14. add-sqr-sqrt 96.5%

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

   15. distribute-neg-in 96.5%

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

   16. add-sqr-sqrt 42.4%

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

   17. sqrt-unprod 69.6%

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

   18. sqr-neg 69.6%

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

   19. sqrt-unprod 33.1%

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

   20. add-sqr-sqrt 63.3%

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

   21. sub-neg 63.3%

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

5. Applied egg-rr 63.3%

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

6. Taylor expanded in t1 around inf 14.8%

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

7. Final simplification 14.8%

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

Reproduce

herbie shell --seed 2023314 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))