2cos (problem 3.3.5)

Average Accuracy: 38.0% → 99.5%

Time: 20.9s

Precision: binary64

Cost: 39104

Math

\[\cos \left(x + \varepsilon\right) - \cos x \]

↓

\[\mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\right)\right) \]

FPCore

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))

↓

(FPCore (x eps)
 :precision binary64
 (fma (- (sin x)) (sin eps) (* (cos x) (* (tan (/ eps 2.0)) (- (sin eps))))))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

↓

double code(double x, double eps) {
	return fma(-sin(x), sin(eps), (cos(x) * (tan((eps / 2.0)) * -sin(eps))));
}

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

↓

function code(x, eps)
	return fma(Float64(-sin(x)), sin(eps), Float64(cos(x) * Float64(tan(Float64(eps / 2.0)) * Float64(-sin(eps)))))
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := N[((-N[Sin[x], $MachinePrecision]) * N[Sin[eps], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.1%

   \[\cos \left(x + \varepsilon\right) - \cos x \]

2. Applied egg-rr 59.0%

   \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
   Step-by-step derivation

   [Start] 36.1	\[ \cos \left(x + \varepsilon\right) - \cos x \]
   cos-sum [=>] 59.0	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]
   sub-neg [=>] 59.0	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]

3. Simplified 59.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos \varepsilon \cdot \cos x\right)} - \cos x \]
   Step-by-step derivation

   [Start] 59.0	\[ \left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
   +-commutative [=>] 59.0	\[ \color{blue}{\left(\left(-\sin x \cdot \sin \varepsilon\right) + \cos x \cdot \cos \varepsilon\right)} - \cos x \]
   distribute-lft-neg-in [=>] 59.0	\[ \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \cos x \cdot \cos \varepsilon\right) - \cos x \]
   *-commutative [=>] 59.0	\[ \left(\color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} + \cos x \cdot \cos \varepsilon\right) - \cos x \]
   fma-def [=>] 59.0	\[ \color{blue}{\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \cos \varepsilon\right)} - \cos x \]
   *-commutative [=>] 59.0	\[ \mathsf{fma}\left(\sin \varepsilon, -\sin x, \color{blue}{\cos \varepsilon \cdot \cos x}\right) - \cos x \]

4. Applied egg-rr 91.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon \cdot -1, \sin x, \cos \varepsilon \cdot \cos x - \cos x\right)} \]
   Step-by-step derivation

   [Start] 59.0	\[ \mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos \varepsilon \cdot \cos x\right) - \cos x \]
   fma-udef [=>] 59.0	\[ \color{blue}{\left(\sin \varepsilon \cdot \left(-\sin x\right) + \cos \varepsilon \cdot \cos x\right)} - \cos x \]
   associate--l+ [=>] 91.3	\[ \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right) + \left(\cos \varepsilon \cdot \cos x - \cos x\right)} \]
   neg-mul-1 [=>] 91.3	\[ \sin \varepsilon \cdot \color{blue}{\left(-1 \cdot \sin x\right)} + \left(\cos \varepsilon \cdot \cos x - \cos x\right) \]
   associate-*r* [=>] 91.3	\[ \color{blue}{\left(\sin \varepsilon \cdot -1\right) \cdot \sin x} + \left(\cos \varepsilon \cdot \cos x - \cos x\right) \]
   fma-def [=>] 91.4	\[ \color{blue}{\mathsf{fma}\left(\sin \varepsilon \cdot -1, \sin x, \cos \varepsilon \cdot \cos x - \cos x\right)} \]

5. Taylor expanded in eps around inf 59.0%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos x \cdot \cos \varepsilon\right) - \cos x} \]

6. Simplified 91.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)} \]
   Step-by-step derivation

   [Start] 59.0	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos x \cdot \cos \varepsilon\right) - \cos x \]
   associate--l+ [=>] 91.3	\[ \color{blue}{-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon - \cos x\right)} \]
   associate-*r* [=>] 91.3	\[ \color{blue}{\left(-1 \cdot \sin x\right) \cdot \sin \varepsilon} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]
   sub-neg [=>] 91.3	\[ \left(-1 \cdot \sin x\right) \cdot \sin \varepsilon + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right)} \]
   *-commutative [=>] 91.3	\[ \left(-1 \cdot \sin x\right) \cdot \sin \varepsilon + \left(\color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) \]
   neg-mul-1 [=>] 91.3	\[ \left(-1 \cdot \sin x\right) \cdot \sin \varepsilon + \left(\cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) \]
   distribute-rgt-in [<=] 91.3	\[ \left(-1 \cdot \sin x\right) \cdot \sin \varepsilon + \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)} \]
   metadata-eval [<=] 91.3	\[ \left(-1 \cdot \sin x\right) \cdot \sin \varepsilon + \cos x \cdot \left(\cos \varepsilon + \color{blue}{\left(-1\right)}\right) \]
   sub-neg [<=] 91.3	\[ \left(-1 \cdot \sin x\right) \cdot \sin \varepsilon + \cos x \cdot \color{blue}{\left(\cos \varepsilon - 1\right)} \]
   fma-udef [<=] 91.4	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \sin x, \sin \varepsilon, \cos x \cdot \left(\cos \varepsilon - 1\right)\right)} \]
   mul-1-neg [=>] 91.4	\[ \mathsf{fma}\left(\color{blue}{-\sin x}, \sin \varepsilon, \cos x \cdot \left(\cos \varepsilon - 1\right)\right) \]
   sub-neg [=>] 91.4	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \color{blue}{\left(\cos \varepsilon + \left(-1\right)\right)}\right) \]
   metadata-eval [=>] 91.4	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + \color{blue}{-1}\right)\right) \]

7. Applied egg-rr 99.1%

   \[\leadsto \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \color{blue}{\left(\left(-{\sin \varepsilon}^{2}\right) \cdot \frac{1}{\cos \varepsilon + 1}\right)}\right) \]
   Step-by-step derivation

   [Start] 91.4	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right) \]
   flip-+ [=>] 91.0	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - -1 \cdot -1}{\cos \varepsilon - -1}}\right) \]
   div-inv [=>] 91.0	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \color{blue}{\left(\left(\cos \varepsilon \cdot \cos \varepsilon - -1 \cdot -1\right) \cdot \frac{1}{\cos \varepsilon - -1}\right)}\right) \]
   metadata-eval [=>] 91.0	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\left(\cos \varepsilon \cdot \cos \varepsilon - \color{blue}{1}\right) \cdot \frac{1}{\cos \varepsilon - -1}\right)\right) \]
   sub-1-cos [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\color{blue}{\left(-\sin \varepsilon \cdot \sin \varepsilon\right)} \cdot \frac{1}{\cos \varepsilon - -1}\right)\right) \]
   pow2 [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\left(-\color{blue}{{\sin \varepsilon}^{2}}\right) \cdot \frac{1}{\cos \varepsilon - -1}\right)\right) \]
   sub-neg [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\left(-{\sin \varepsilon}^{2}\right) \cdot \frac{1}{\color{blue}{\cos \varepsilon + \left(--1\right)}}\right)\right) \]
   metadata-eval [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\left(-{\sin \varepsilon}^{2}\right) \cdot \frac{1}{\cos \varepsilon + \color{blue}{1}}\right)\right) \]

8. Simplified 99.6%

   \[\leadsto \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \color{blue}{\left(-\tan \left(\frac{\varepsilon}{2}\right) \cdot \sin \varepsilon\right)}\right) \]
   Step-by-step derivation

   [Start] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\left(-{\sin \varepsilon}^{2}\right) \cdot \frac{1}{\cos \varepsilon + 1}\right)\right) \]
   associate-*r/ [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \color{blue}{\frac{\left(-{\sin \varepsilon}^{2}\right) \cdot 1}{\cos \varepsilon + 1}}\right) \]
   *-rgt-identity [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \frac{\color{blue}{-{\sin \varepsilon}^{2}}}{\cos \varepsilon + 1}\right) \]
   distribute-frac-neg [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \color{blue}{\left(-\frac{{\sin \varepsilon}^{2}}{\cos \varepsilon + 1}\right)}\right) \]
   unpow2 [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(-\frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{\cos \varepsilon + 1}\right)\right) \]
   associate-/l* [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(-\color{blue}{\frac{\sin \varepsilon}{\frac{\cos \varepsilon + 1}{\sin \varepsilon}}}\right)\right) \]
   associate-/r/ [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(-\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon + 1} \cdot \sin \varepsilon}\right)\right) \]
   +-commutative [=>] 99.1	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(-\frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}} \cdot \sin \varepsilon\right)\right) \]
   hang-0p-tan [=>] 99.6	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(-\color{blue}{\tan \left(\frac{\varepsilon}{2}\right)} \cdot \sin \varepsilon\right)\right) \]

9. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	39176

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.75 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-5}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sin \varepsilon, \sin x, \cos x \cdot \cos \varepsilon - \cos x\right)\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	32777

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.3 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.9 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \end{array} \]

Alternative 3
Accuracy	99.2%
Cost	32777

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.75 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \end{array} \]

Alternative 5
Accuracy	76.3%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0112 \lor \neg \left(\varepsilon \leq 0.0085\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \end{array} \]

Alternative 6
Accuracy	76.3%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.011 \lor \neg \left(\varepsilon \leq 0.0062\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\\ \end{array} \]

Alternative 7
Accuracy	75.7%
Cost	13632

\[-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

Alternative 8
Accuracy	76.1%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0077 \lor \neg \left(\varepsilon \leq 0.00095\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - \sin x\right)\\ \end{array} \]

Alternative 9
Accuracy	75.4%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0073 \lor \neg \left(\varepsilon \leq 0.0145\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - \sin x\right)\\ \end{array} \]

Alternative 10
Accuracy	65.9%
Cost	7052

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.35 \cdot 10^{-94}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	52.4%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000112 \lor \neg \left(\varepsilon \leq 0.00013\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - x \cdot \varepsilon\\ \end{array} \]

Alternative 12
Accuracy	23.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-115} \lor \neg \left(x \leq 1.32 \cdot 10^{-80}\right):\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 13
Accuracy	25.9%
Cost	576

\[-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - x \cdot \varepsilon \]

Alternative 14
Accuracy	25.9%
Cost	448

\[\varepsilon \cdot \left(\varepsilon \cdot -0.5 - x\right) \]

Alternative 15
Accuracy	17.3%
Cost	256

\[x \cdot \left(-\varepsilon\right) \]

Reproduce

herbie shell --seed 2023157 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))