Rosa's DopplerBench

Percentage Accurate: 73.3% → 98.2%

Time: 10.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. times-frac 98.3%

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

   2. distribute-frac-neg 98.3%

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

   3. distribute-neg-frac2 98.3%

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

   4. +-commutative 98.3%

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

   5. distribute-neg-in 98.3%

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

   6. unsub-neg 98.3%

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

3. Simplified 98.3%

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

5. Final simplification 98.3%

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

Alternative 2: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := \frac{v}{t1 + u}\\ t_3 := t1 \cdot \frac{t\_2}{t\_1}\\ \mathbf{if}\;t1 \leq -1.16 \cdot 10^{+156}:\\ \;\;\;\;t\_2 \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{elif}\;t1 \leq -4.8 \cdot 10^{-126}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t1 \leq -1.15 \cdot 10^{-307}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t\_1}\\ \mathbf{elif}\;t1 \leq 1.12 \cdot 10^{+105}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (/ v (+ t1 u))) (t_3 (* t1 (/ t_2 t_1))))
   (if (<= t1 -1.16e+156)
     (* t_2 (+ (/ u t1) -1.0))
     (if (<= t1 -4.8e-126)
       t_3
       (if (<= t1 -1.15e-307)
         (/ (/ t1 (/ u v)) t_1)
         (if (<= t1 1.12e+105) t_3 (/ v t_1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v / (t1 + u);
	double t_3 = t1 * (t_2 / t_1);
	double tmp;
	if (t1 <= -1.16e+156) {
		tmp = t_2 * ((u / t1) + -1.0);
	} else if (t1 <= -4.8e-126) {
		tmp = t_3;
	} else if (t1 <= -1.15e-307) {
		tmp = (t1 / (u / v)) / t_1;
	} else if (t1 <= 1.12e+105) {
		tmp = t_3;
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = -u - t1
    t_2 = v / (t1 + u)
    t_3 = t1 * (t_2 / t_1)
    if (t1 <= (-1.16d+156)) then
        tmp = t_2 * ((u / t1) + (-1.0d0))
    else if (t1 <= (-4.8d-126)) then
        tmp = t_3
    else if (t1 <= (-1.15d-307)) then
        tmp = (t1 / (u / v)) / t_1
    else if (t1 <= 1.12d+105) then
        tmp = t_3
    else
        tmp = v / t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v / (t1 + u);
	double t_3 = t1 * (t_2 / t_1);
	double tmp;
	if (t1 <= -1.16e+156) {
		tmp = t_2 * ((u / t1) + -1.0);
	} else if (t1 <= -4.8e-126) {
		tmp = t_3;
	} else if (t1 <= -1.15e-307) {
		tmp = (t1 / (u / v)) / t_1;
	} else if (t1 <= 1.12e+105) {
		tmp = t_3;
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	t_2 = v / (t1 + u)
	t_3 = t1 * (t_2 / t_1)
	tmp = 0
	if t1 <= -1.16e+156:
		tmp = t_2 * ((u / t1) + -1.0)
	elif t1 <= -4.8e-126:
		tmp = t_3
	elif t1 <= -1.15e-307:
		tmp = (t1 / (u / v)) / t_1
	elif t1 <= 1.12e+105:
		tmp = t_3
	else:
		tmp = v / t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	t_2 = Float64(v / Float64(t1 + u))
	t_3 = Float64(t1 * Float64(t_2 / t_1))
	tmp = 0.0
	if (t1 <= -1.16e+156)
		tmp = Float64(t_2 * Float64(Float64(u / t1) + -1.0));
	elseif (t1 <= -4.8e-126)
		tmp = t_3;
	elseif (t1 <= -1.15e-307)
		tmp = Float64(Float64(t1 / Float64(u / v)) / t_1);
	elseif (t1 <= 1.12e+105)
		tmp = t_3;
	else
		tmp = Float64(v / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	t_2 = v / (t1 + u);
	t_3 = t1 * (t_2 / t_1);
	tmp = 0.0;
	if (t1 <= -1.16e+156)
		tmp = t_2 * ((u / t1) + -1.0);
	elseif (t1 <= -4.8e-126)
		tmp = t_3;
	elseif (t1 <= -1.15e-307)
		tmp = (t1 / (u / v)) / t_1;
	elseif (t1 <= 1.12e+105)
		tmp = t_3;
	else
		tmp = v / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, Block[{t$95$2 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t1 * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.16e+156], N[(t$95$2 * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -4.8e-126], t$95$3, If[LessEqual[t1, -1.15e-307], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t1, 1.12e+105], t$95$3, N[(v / t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.16e156

   1. Initial program 32.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 91.5%

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

   if -1.16e156 < t1 < -4.80000000000000014e-126 or -1.1499999999999999e-307 < t1 < 1.12e105

   1. Initial program 83.0%

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

      1. associate-/l* 85.5%

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

      2. distribute-lft-neg-out 85.5%

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

      3. distribute-rgt-neg-in 85.5%

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

      4. associate-/r* 93.8%

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

      5. distribute-neg-frac2 93.8%

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

   3. Simplified 93.8%

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

   if -4.80000000000000014e-126 < t1 < -1.1499999999999999e-307

   1. Initial program 85.1%

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

      1. associate-/l* 75.4%

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

      2. distribute-lft-neg-out 75.4%

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

      3. distribute-rgt-neg-in 75.4%

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

      4. associate-/r* 79.9%

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

      5. distribute-neg-frac2 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in t1 around 0 77.4%

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

      1. clear-num 77.4%

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

      2. un-div-inv 77.5%

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

      3. div-inv 77.2%

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

      4. add-sqr-sqrt 46.6%

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

      5. sqrt-unprod 58.8%

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

      6. sqr-neg 58.8%

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

      7. sqrt-unprod 16.6%

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

      8. add-sqr-sqrt 37.8%

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

      9. clear-num 37.8%

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

   7. Applied egg-rr 37.8%

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

      1. frac-2neg 37.8%

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

      2. distribute-frac-neg2 37.8%

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

      3. add-sqr-sqrt 37.8%

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

      4. sqrt-unprod 35.7%

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

      5. sqr-neg 35.7%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 77.4%

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

      8. *-commutative 77.4%

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

      9. associate-/r* 92.0%

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

   9. Applied egg-rr 92.0%

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

   if 1.12e105 < t1

   1. Initial program 53.3%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 93.5%

      \[\leadsto \color{blue}{-1} \cdot \frac{v}{t1 + u} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.16 \cdot 10^{+156}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{elif}\;t1 \leq -4.8 \cdot 10^{-126}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -1.15 \cdot 10^{-307}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 1.12 \cdot 10^{+105}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{-60} \lor \neg \left(t1 \leq 3.6 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.35e-60) (not (<= t1 3.6e+71)))
   (/ v (- (- u) t1))
   (* (/ t1 (- u)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.35e-60) || !(t1 <= 3.6e+71)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.35d-60)) .or. (.not. (t1 <= 3.6d+71))) then
        tmp = v / (-u - t1)
    else
        tmp = (t1 / -u) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.35e-60) || !(t1 <= 3.6e+71)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.35e-60) or not (t1 <= 3.6e+71):
		tmp = v / (-u - t1)
	else:
		tmp = (t1 / -u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.35e-60) || !(t1 <= 3.6e+71))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.35e-60) || ~((t1 <= 3.6e+71)))
		tmp = v / (-u - t1);
	else
		tmp = (t1 / -u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.35e-60], N[Not[LessEqual[t1, 3.6e+71]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.35e-60 or 3.6e71 < t1

   1. Initial program 53.0%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 84.8%

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

   if -1.35e-60 < t1 < 3.6e71

   1. Initial program 88.2%

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

      1. times-frac 97.0%

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

      2. distribute-frac-neg 97.0%

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

      3. distribute-neg-frac2 97.0%

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

      4. +-commutative 97.0%

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

      5. distribute-neg-in 97.0%

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

      6. unsub-neg 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in t1 around 0 77.5%

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

      1. associate-*r/ 77.5%

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

      2. mul-1-neg 77.5%

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

   7. Simplified 77.5%

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

   8. Taylor expanded in t1 around 0 80.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{-60} \lor \neg \left(t1 \leq 3.6 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{elif}\;t1 \leq 3.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -3.2e-60)
   (* (/ v (+ t1 u)) (+ (/ u t1) -1.0))
   (if (<= t1 3.7e+71) (* (/ t1 (- u)) (/ v u)) (/ v (- (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.2e-60) {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	} else if (t1 <= 3.7e+71) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-3.2d-60)) then
        tmp = (v / (t1 + u)) * ((u / t1) + (-1.0d0))
    else if (t1 <= 3.7d+71) then
        tmp = (t1 / -u) * (v / u)
    else
        tmp = v / (-u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.2e-60) {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	} else if (t1 <= 3.7e+71) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -3.2e-60:
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0)
	elif t1 <= 3.7e+71:
		tmp = (t1 / -u) * (v / u)
	else:
		tmp = v / (-u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -3.2e-60)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(u / t1) + -1.0));
	elseif (t1 <= 3.7e+71)
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	else
		tmp = Float64(v / Float64(Float64(-u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -3.2e-60)
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	elseif (t1 <= 3.7e+71)
		tmp = (t1 / -u) * (v / u);
	else
		tmp = v / (-u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -3.2e-60], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.7e+71], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.2000000000000001e-60

   1. Initial program 55.1%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 82.2%

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

   if -3.2000000000000001e-60 < t1 < 3.7e71

   1. Initial program 88.2%

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

      1. times-frac 97.0%

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

      2. distribute-frac-neg 97.0%

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

      3. distribute-neg-frac2 97.0%

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

      4. +-commutative 97.0%

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

      5. distribute-neg-in 97.0%

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

      6. unsub-neg 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in t1 around 0 77.5%

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

      1. associate-*r/ 77.5%

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

      2. mul-1-neg 77.5%

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

   7. Simplified 77.5%

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

   8. Taylor expanded in t1 around 0 80.7%

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

   if 3.7e71 < t1

   1. Initial program 50.1%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 88.6%

      \[\leadsto \color{blue}{-1} \cdot \frac{v}{t1 + u} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{elif}\;t1 \leq 3.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{-139} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-168}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.5e-139) (not (<= t1 3.1e-168)))
   (/ v (- (- u) t1))
   (* v (/ (/ t1 u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.5e-139) || !(t1 <= 3.1e-168)) {
		tmp = v / (-u - t1);
	} else {
		tmp = v * ((t1 / u) / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-9.5d-139)) .or. (.not. (t1 <= 3.1d-168))) then
        tmp = v / (-u - t1)
    else
        tmp = v * ((t1 / u) / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.5e-139) || !(t1 <= 3.1e-168)) {
		tmp = v / (-u - t1);
	} else {
		tmp = v * ((t1 / u) / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9.5e-139) or not (t1 <= 3.1e-168):
		tmp = v / (-u - t1)
	else:
		tmp = v * ((t1 / u) / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9.5e-139) || !(t1 <= 3.1e-168))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(v * Float64(Float64(t1 / u) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -9.5e-139) || ~((t1 <= 3.1e-168)))
		tmp = v / (-u - t1);
	else
		tmp = v * ((t1 / u) / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9.5e-139], N[Not[LessEqual[t1, 3.1e-168]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[(t1 / u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.5000000000000006e-139 or 3.1e-168 < t1

   1. Initial program 67.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 71.4%

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

   if -9.5000000000000006e-139 < t1 < 3.1e-168

   1. Initial program 83.2%

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

      1. associate-/l* 79.7%

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

      2. distribute-lft-neg-out 79.7%

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

      3. distribute-rgt-neg-in 79.7%

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

      4. associate-/r* 83.7%

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

      5. distribute-neg-frac2 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in t1 around 0 80.5%

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

   6. Taylor expanded in u around 0 14.7%

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

      1. mul-1-neg 14.7%

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

      2. associate-/r* 14.7%

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

      3. distribute-neg-frac2 14.7%

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

   8. Simplified 14.7%

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

      1. clear-num 14.7%

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

      2. un-div-inv 14.7%

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

      3. div-inv 14.7%

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

      4. add-sqr-sqrt 7.6%

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

      5. sqrt-unprod 22.3%

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

      6. sqr-neg 22.3%

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

      7. sqrt-unprod 9.2%

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

      8. add-sqr-sqrt 18.0%

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

      9. clear-num 18.0%

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

   10. Applied egg-rr 18.0%

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

       1. associate-/r* 28.1%

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

       2. associate-/r/ 40.7%

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

   12. Applied egg-rr 40.7%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{-139} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-168}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 3.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= u 3.1e+152) (/ v (- t1)) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.1e+152) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 3.1d+152) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.1e+152) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 3.1e+152:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 3.1e+152)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 3.1e+152)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 3.1e+152], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 3.1e152

   1. Initial program 70.1%

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

      1. associate-/l* 70.2%

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

      2. distribute-lft-neg-out 70.2%

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

      3. distribute-rgt-neg-in 70.2%

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

      4. associate-/r* 81.2%

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

      5. distribute-neg-frac2 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in t1 around inf 56.7%

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

      1. associate-*r/ 56.7%

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

      2. neg-mul-1 56.7%

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

   7. Simplified 56.7%

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

   if 3.1e152 < u

   1. Initial program 80.8%

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

      1. associate-/l* 81.0%

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

      2. distribute-lft-neg-out 81.0%

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

      3. distribute-rgt-neg-in 81.0%

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

      4. associate-/r* 86.4%

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

      5. distribute-neg-frac2 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in t1 around 0 86.5%

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

   6. Taylor expanded in u around 0 51.0%

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

      1. mul-1-neg 51.0%

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

      2. associate-/r* 50.7%

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

      3. distribute-neg-frac2 50.7%

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

   8. Simplified 50.7%

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

      1. clear-num 50.7%

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

      2. un-div-inv 50.7%

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

      3. div-inv 50.7%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 64.0%

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

      6. sqr-neg 64.0%

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

      7. sqrt-unprod 50.6%

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

      8. add-sqr-sqrt 50.6%

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

      9. clear-num 50.6%

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

   10. Applied egg-rr 50.6%

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

   11. Taylor expanded in t1 around 0 40.6%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 3.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(u, v, t1)
	return Float64(v / Float64(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 71.6%

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

   1. times-frac 98.3%

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

   2. distribute-frac-neg 98.3%

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

   3. distribute-neg-frac2 98.3%

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

   4. +-commutative 98.3%

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

   5. distribute-neg-in 98.3%

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

   6. unsub-neg 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in t1 around inf 58.7%

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

6. Final simplification 58.7%

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

Alternative 8: 17.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. associate-/l* 71.7%

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

   2. distribute-lft-neg-out 71.7%

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

   3. distribute-rgt-neg-in 71.7%

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

   4. associate-/r* 81.9%

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

   5. distribute-neg-frac2 81.9%

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

3. Simplified 81.9%

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

5. Taylor expanded in t1 around 0 53.5%

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

6. Taylor expanded in u around 0 20.7%

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

   1. mul-1-neg 20.7%

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

   2. associate-/r* 22.3%

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

   3. distribute-neg-frac2 22.3%

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

8. Simplified 22.3%

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

   1. clear-num 22.3%

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

   2. un-div-inv 22.3%

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

   3. div-inv 22.5%

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

   4. add-sqr-sqrt 10.3%

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

   5. sqrt-unprod 26.7%

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

   6. sqr-neg 26.7%

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

   7. sqrt-unprod 12.8%

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

   8. add-sqr-sqrt 23.5%

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

   9. clear-num 23.5%

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

10. Applied egg-rr 23.5%

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

11. Taylor expanded in t1 around 0 16.6%

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

12. Final simplification 16.6%

    \[\leadsto \frac{v}{u} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024054 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))