Rosa's DopplerBench

Average Accuracy: 71.0% → 97.9%

Time: 12.3s

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 71.0%

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

2. Simplified 97.9%

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

   [Start] 71.0	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 97.9	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 97.9%

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

Alternatives

Alternative 1
Accuracy	73.4%
Cost	1172

\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -8.8 \cdot 10^{+112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -5.2 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -7.8 \cdot 10^{-78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -6.2 \cdot 10^{-92}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{elif}\;t1 \leq 3.1 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	73.6%
Cost	1041

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -8.8 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -7.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{elif}\;t1 \leq -8.4 \cdot 10^{-78} \lor \neg \left(t1 \leq 2.2 \cdot 10^{-127}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 3
Accuracy	73.5%
Cost	1041

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -8.8 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -9.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq -7 \cdot 10^{-78} \lor \neg \left(t1 \leq 7.6 \cdot 10^{-126}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 4
Accuracy	77.7%
Cost	905

\[\begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{-40} \lor \neg \left(u \leq 3400000000\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 5
Accuracy	77.4%
Cost	904

\[\begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 8.6:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{t1 + u}{\frac{v}{u}}}\\ \end{array} \]

Alternative 6
Accuracy	95.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+196}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \end{array} \]

Alternative 7
Accuracy	95.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;u \leq -1.38 \cdot 10^{+197}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2 - \frac{u}{t1}\right) - t1}\\ \end{array} \]

Alternative 8
Accuracy	75.5%
Cost	777

\[\begin{array}{l} \mathbf{if}\;u \leq -1.26 \cdot 10^{-40} \lor \neg \left(u \leq 55000000000\right):\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 9
Accuracy	65.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{+142} \lor \neg \left(u \leq 1.08 \cdot 10^{+220}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 10
Accuracy	66.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -2.25 \cdot 10^{+145} \lor \neg \left(u \leq 6.5 \cdot 10^{+165}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 11
Accuracy	67.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+145} \lor \neg \left(u \leq 7 \cdot 10^{+164}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 12
Accuracy	97.7%
Cost	704

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

Alternative 13
Accuracy	56.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{+218} \lor \neg \left(u \leq 1.95 \cdot 10^{+223}\right):\\ \;\;\;\;\frac{v}{\frac{u}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 14
Accuracy	56.3%
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{+218} \lor \neg \left(u \leq 2.5 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 15
Accuracy	60.8%
Cost	384

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

Alternative 16
Accuracy	52.7%
Cost	256

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

Reproduce

herbie shell --seed 2023143 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))