Rosa's DopplerBench

Average Accuracy: 71.9% → 97.8%

Time: 15.0s

Precision: binary64

Cost: 768

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 71.9%

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

2. Simplified 97.8%

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

   [Start] 71.9	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 97.8	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 97.8%

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

Alternatives

Alternative 1
Accuracy	75.7%
Cost	1305

\[\begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{-1}\\ t_2 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -3.8 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -3.8 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.75 \cdot 10^{-89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 4 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 8 \cdot 10^{+83} \lor \neg \left(u \leq 2.1 \cdot 10^{+102}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 2
Accuracy	76.8%
Cost	1104

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ t_2 := \frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \mathbf{if}\;u \leq -6.1 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -3.8 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.75 \cdot 10^{-89}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	76.8%
Cost	1104

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;u \leq -1.32 \cdot 10^{+31}:\\ \;\;\;\;\frac{t1}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq -3.8 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.75 \cdot 10^{-89}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 4
Accuracy	78.0%
Cost	1041

\[\begin{array}{l} t_1 := \frac{-v}{u}\\ t_2 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;t1 \leq -3.1 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\frac{t_1}{\frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq -1.7 \cdot 10^{-52} \lor \neg \left(t1 \leq 5.8 \cdot 10^{-96}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot t_1\\ \end{array} \]

Alternative 5
Accuracy	76.0%
Cost	1040

\[\begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{-1}\\ t_2 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -1.2 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -3.8 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.75 \cdot 10^{-89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]

Alternative 6
Accuracy	76.0%
Cost	1040

\[\begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{if}\;u \leq -3.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq -4.5 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.75 \cdot 10^{-89}:\\ \;\;\;\;\frac{-v}{\frac{u}{\frac{t1}{u}}}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]

Alternative 7
Accuracy	76.0%
Cost	1040

\[\begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{if}\;u \leq -2.7 \cdot 10^{+43}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq -3.8 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.75 \cdot 10^{-89}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]

Alternative 8
Accuracy	75.9%
Cost	1040

\[\begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{if}\;u \leq -1.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq -4.4 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.75 \cdot 10^{-89}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{elif}\;u \leq 5.3 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \]

Alternative 9
Accuracy	58.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+111} \lor \neg \left(u \leq 1.75 \cdot 10^{+127}\right):\\ \;\;\;\;v \cdot \frac{t1}{t1 \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10
Accuracy	68.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{+64} \lor \neg \left(u \leq 8.2 \cdot 10^{+102}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11
Accuracy	68.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{+117} \lor \neg \left(u \leq 1.25 \cdot 10^{+112}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \end{array} \]

Alternative 12
Accuracy	97.6%
Cost	704

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

Alternative 13
Accuracy	54.0%
Cost	521

\[\begin{array}{l} \mathbf{if}\;t1 \leq -4.1 \cdot 10^{-173} \lor \neg \left(t1 \leq 4.9 \cdot 10^{-206}\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 14
Accuracy	23.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 4.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 15
Accuracy	14.7%
Cost	192

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

Reproduce

herbie shell --seed 2023126 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))