Rosa's DopplerBench

Average Accuracy: 71.3% → 98.2%

Time: 11.2s

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(t1 / Float64(t1 + u)) * Float64(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 71.3%

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

2. Simplified 98.2%

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

   [Start] 71.3	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 98.2	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 98.2%

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

Alternatives

Alternative 1
Accuracy	71.2%
Cost	1042

\[\begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{-136} \lor \neg \left(t1 \leq 1.12 \cdot 10^{-98} \lor \neg \left(t1 \leq 3.85 \cdot 10^{+21}\right) \land t1 \leq 4.8 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \]

Alternative 2
Accuracy	74.9%
Cost	777

\[\begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+58} \lor \neg \left(u \leq 0.0061\right):\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 3
Accuracy	75.9%
Cost	777

\[\begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+58} \lor \neg \left(u \leq 0.17\right):\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]

Alternative 4
Accuracy	75.0%
Cost	776

\[\begin{array}{l} \mathbf{if}\;u \leq -1.65 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 0.0001:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 5
Accuracy	75.4%
Cost	776

\[\begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 0.00052:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 6
Accuracy	65.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+67} \lor \neg \left(u \leq 7 \cdot 10^{+100}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 7
Accuracy	66.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -9.2 \cdot 10^{+67} \lor \neg \left(u \leq 1.05 \cdot 10^{+101}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8
Accuracy	98.1%
Cost	704

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

Alternative 9
Accuracy	55.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+58} \lor \neg \left(u \leq 3.2 \cdot 10^{+108}\right):\\ \;\;\;\;-0.5 \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10
Accuracy	55.7%
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -2.55 \cdot 10^{+58} \lor \neg \left(u \leq 5.2 \cdot 10^{+108}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11
Accuracy	52.9%
Cost	256

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

Alternative 12
Accuracy	14.2%
Cost	192

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

Reproduce

herbie shell --seed 2023135 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))