Linear.Quaternion:$ccos from linear-1.19.1.3

Average Accuracy: 100.0% → 100.0%

Time: 12.4s

Precision: binary64

Cost: 13120

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\sin x \cdot \frac{\sinh y}{y} \]

↓

\[\frac{\sin x}{\frac{y}{\sinh y}} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) y)))

↓

(FPCore (x y) :precision binary64 (/ (sin x) (/ y (sinh y))))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / y);
}

↓

double code(double x, double y) {
	return sin(x) / (y / sinh(y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) / (y / sinh(y))
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / y);
}

↓

public static double code(double x, double y) {
	return Math.sin(x) / (y / Math.sinh(y));
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / y)

↓

def code(x, y):
	return math.sin(x) / (y / math.sinh(y))

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / y))
end

↓

function code(x, y)
	return Float64(sin(x) / Float64(y / sinh(y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / y);
end

↓

function tmp = code(x, y)
	tmp = sin(x) / (y / sinh(y));
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]

2. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\sin x}{\frac{y}{\sinh y}}} \]
   Proof

   [Start] 100.0	\[ \sin x \cdot \frac{\sinh y}{y} \]
   clear-num [=>] 100.0	\[ \sin x \cdot \color{blue}{\frac{1}{\frac{y}{\sinh y}}} \]
   un-div-inv [=>] 100.0	\[ \color{blue}{\frac{\sin x}{\frac{y}{\sinh y}}} \]

3. Final simplification 100.0%

   \[\leadsto \frac{\sin x}{\frac{y}{\sinh y}} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13120

\[\sin x \cdot \frac{\sinh y}{y} \]

Alternative 2
Accuracy	83.8%
Cost	7688

\[\begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := \sin x \cdot t_0\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \frac{1 - {y}^{4} \cdot 0.027777777777777776}{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq -680:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left(y \cdot y\right) \cdot \left(x + -0.16666666666666666 \cdot {x}^{3}\right)\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+18}:\\ \;\;\;\;\sin x \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{t_0 \cdot t_0 + -1}{t_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	83.9%
Cost	7564

\[\begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := \sin x \cdot t_0\\ t_2 := x \cdot \frac{t_0 \cdot t_0 + -1}{t_0 + -1}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{+113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -440:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left(y \cdot y\right) \cdot \left(x + -0.16666666666666666 \cdot {x}^{3}\right)\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+19}:\\ \;\;\;\;\sin x \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	79.3%
Cost	7376

\[\begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := 0.16666666666666666 \cdot \left(y \cdot \left(\sin x \cdot y\right)\right)\\ t_2 := x \cdot \frac{t_0 \cdot t_0 + -1}{t_0 + -1}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 0.0025:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	84.8%
Cost	7376

\[\begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := \sin x \cdot t_0\\ t_2 := x \cdot \frac{t_0 \cdot t_0 + -1}{t_0 + -1}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 0.0025:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	84.8%
Cost	7376

\[\begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := \sin x \cdot t_0\\ t_2 := x \cdot \frac{t_0 \cdot t_0 + -1}{t_0 + -1}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\sin x \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	78.6%
Cost	6860

\[\begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := x \cdot \frac{t_0 \cdot t_0 + -1}{t_0 + -1}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq -2.85 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.0027:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot t_0\\ \end{array} \]

Alternative 8
Accuracy	54.9%
Cost	1736

\[\begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+150}:\\ \;\;\;\;x \cdot \frac{t_0 \cdot t_0 + -1}{t_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot t_0\\ \end{array} \]

Alternative 9
Accuracy	48.2%
Cost	832

\[x \cdot \left(\left(1 + \left(0.16666666666666666 \cdot \left(y \cdot y\right) + 1\right)\right) + -1\right) \]

Alternative 10
Accuracy	42.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-12} \lor \neg \left(y \leq 5.8 \cdot 10^{+14}\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	47.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-12} \lor \neg \left(y \leq 5.8 \cdot 10^{+14}\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Accuracy	47.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right) + -1\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	45.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 2000000:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(y \cdot \left(x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	48.2%
Cost	576

\[x + x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

Alternative 15
Accuracy	26.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023160 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))