Rosa's DopplerBench

Average Error: 18.4 → 1.3

Time: 14.3s

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 18.4

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

2. Simplified 1.3

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

   [Start] 18.4	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 1.3	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 1.3

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

Alternatives

Alternative 1
Error	14.1
Cost	1232

\[\begin{array}{l} t_1 := \frac{v}{t1 + u}\\ t_2 := \frac{t_1}{\frac{-u}{t1}}\\ \mathbf{if}\;u \leq -9.2 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{-52}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 9.8 \cdot 10^{+76}:\\ \;\;\;\;t_1 \cdot \left(-1 + \frac{u}{t1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \left(-\frac{v}{u}\right)}{u}\\ \end{array} \]

Alternative 2
Error	14.1
Cost	1232

\[\begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{\frac{-u}{t1}}\\ \mathbf{if}\;u \leq -1.15 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{\frac{u \cdot 2}{v} + \frac{t1}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \left(-\frac{v}{u}\right)}{u}\\ \end{array} \]

Alternative 3
Error	14.0
Cost	1232

\[\begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{\frac{-u}{t1}}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{t1}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{\frac{u \cdot 2}{v} + \frac{t1}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \left(-\frac{v}{u}\right)}{u}\\ \end{array} \]

Alternative 4
Error	15.4
Cost	1040

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -7.6 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -2.55 \cdot 10^{-79}:\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq -9 \cdot 10^{-102}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{-155}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.5
Cost	1040

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -2.7 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -3.3 \cdot 10^{-80}:\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq -1.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	14.2
Cost	1040

\[\begin{array}{l} t_1 := -\frac{v}{u}\\ t_2 := t1 \cdot \frac{t_1}{t1 + u}\\ \mathbf{if}\;u \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 3.25 \cdot 10^{-42}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4.2 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+87}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot t_1}{u}\\ \end{array} \]

Alternative 7
Error	13.9
Cost	1040

\[\begin{array}{l} t_1 := -\frac{v}{u}\\ \mathbf{if}\;u \leq -1.85 \cdot 10^{-28}:\\ \;\;\;\;\frac{-v}{t1 + u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 1.22 \cdot 10^{-42}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{+72}:\\ \;\;\;\;t1 \cdot \frac{t_1}{t1 + u}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot t_1}{u}\\ \end{array} \]

Alternative 8
Error	14.0
Cost	1040

\[\begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{\frac{-u}{t1}}\\ \mathbf{if}\;u \leq -9.6 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{+89}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \left(-\frac{v}{u}\right)}{u}\\ \end{array} \]

Alternative 9
Error	15.6
Cost	777

\[\begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{-28} \lor \neg \left(u \leq 2.5 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 10
Error	15.7
Cost	777

\[\begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{-28} \lor \neg \left(u \leq 5.6 \cdot 10^{+92}\right):\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 11
Error	15.1
Cost	777

\[\begin{array}{l} \mathbf{if}\;u \leq -8.6 \cdot 10^{-29} \lor \neg \left(u \leq 2.2 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{t1 \cdot \left(-\frac{v}{u}\right)}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 12
Error	21.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+154} \lor \neg \left(u \leq 5.5 \cdot 10^{+92}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 13
Error	20.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -3.3 \cdot 10^{+74} \lor \neg \left(u \leq 6.2 \cdot 10^{+91}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 14
Error	20.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+74}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 4.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \end{array} \]

Alternative 15
Error	1.3
Cost	704

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

Alternative 16
Error	27.7
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+148} \lor \neg \left(u \leq 4.6 \cdot 10^{+141}\right):\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 17
Error	25.3
Cost	384

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

Alternative 18
Error	30.8
Cost	256

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

Reproduce

herbie shell --seed 2023066 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))