Rosa's DopplerBench

Average Accuracy: 71.9% → 98.1%

Time: 11.1s

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 98.1%

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

   [Start] 71.9	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 98.1	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 98.1%

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

Alternatives

Alternative 1
Accuracy	77.1%
Cost	1173

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.95 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{-257}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{-129} \lor \neg \left(t1 \leq 2.7 \cdot 10^{-32}\right) \land t1 \leq 7.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	76.5%
Cost	1042

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3.7 \cdot 10^{-91} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-129} \lor \neg \left(t1 \leq 2.15 \cdot 10^{-32}\right) \land t1 \leq 3.1 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \]

Alternative 3
Accuracy	77.6%
Cost	1042

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-91} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-129} \lor \neg \left(t1 \leq 2.8 \cdot 10^{-32}\right) \land t1 \leq 5.5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ \end{array} \]

Alternative 4
Accuracy	78.0%
Cost	1040

\[\begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -4 \cdot 10^{-91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 1.7 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 7.8 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	75.0%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3.6 \cdot 10^{-107} \lor \neg \left(t1 \leq 6 \cdot 10^{-130}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u \cdot u}{v}}\\ \end{array} \]

Alternative 6
Accuracy	68.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+149} \lor \neg \left(u \leq 2.55 \cdot 10^{+120}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 7
Accuracy	68.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 8
Accuracy	68.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -1.12 \cdot 10^{+150}:\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \mathbf{elif}\;u \leq 8 \cdot 10^{+121}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 9
Accuracy	94.8%
Cost	704

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

Alternative 10
Accuracy	97.9%
Cost	704

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

Alternative 11
Accuracy	57.3%
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{+153} \lor \neg \left(u \leq 8.6 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 12
Accuracy	61.6%
Cost	384

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

Alternative 13
Accuracy	52.7%
Cost	256

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

Reproduce

herbie shell --seed 2023140 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))