Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.5 → 0.5

Time: 11.5s

Precision: binary64

Cost: 20680

Math

\[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

↓

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;t_0 \leq 10^{-34}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{x}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (<= t_0 (- INFINITY))
     (/ (/ y z) x)
     (if (<= t_0 1e-34)
       (/ (/ (* (cosh x) y) x) z)
       (/ (* (/ y z) (+ 1.0 (* 0.5 (* x x)))) x)))))

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}

↓

double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y / z) / x;
	} else if (t_0 <= 1e-34) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = ((y / z) * (1.0 + (0.5 * (x * x)))) / x;
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (Math.cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (y / z) / x;
	} else if (t_0 <= 1e-34) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = ((y / z) * (1.0 + (0.5 * (x * x)))) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (math.cosh(x) * (y / x)) / z

↓

def code(x, y, z):
	t_0 = (math.cosh(x) * (y / x)) / z
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (y / z) / x
	elif t_0 <= 1e-34:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = ((y / z) * (1.0 + (0.5 * (x * x)))) / x
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(cosh(x) * Float64(y / x)) / z)
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(cosh(x) * Float64(y / x)) / z)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y / z) / x);
	elseif (t_0 <= 1e-34)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = Float64(Float64(Float64(y / z) * Float64(1.0 + Float64(0.5 * Float64(x * x)))) / x);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (cosh(x) * (y / x)) / z;
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (cosh(x) * (y / x)) / z;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (y / z) / x;
	elseif (t_0 <= 1e-34)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = ((y / z) * (1.0 + (0.5 * (x * x)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-34], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Target

Original	7.5
Target	0.4
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -inf.0

   1. Initial program 64.0

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Simplified 0.6

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]
      Proof
      (*.f64 y (/.f64 (/.f64 (cosh.f64 x) z) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (/.f64 (cosh.f64 x) z)) x)): 60 points increase in error, 68 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 y x) (/.f64 (cosh.f64 x) z))): 70 points increase in error, 62 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (cosh.f64 x) z) (/.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)): 17 points increase in error, 41 points decrease in error

   3. Taylor expanded in x around 0 0.9

      \[\leadsto y \cdot \frac{\color{blue}{\frac{1}{z}}}{x} \]

   4. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if -inf.0 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.99999999999999928e-35

   1. Initial program 0.2

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Applied egg-rr 0.2

      \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

   if 9.99999999999999928e-35 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 11.4

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Simplified 12.0

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]
      Proof
      (*.f64 y (/.f64 (/.f64 (cosh.f64 x) z) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (/.f64 (cosh.f64 x) z)) x)): 60 points increase in error, 68 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 y x) (/.f64 (cosh.f64 x) z))): 70 points increase in error, 62 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (cosh.f64 x) z) (/.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)): 17 points increase in error, 41 points decrease in error

   3. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

   4. Taylor expanded in x around 0 1.0

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{y \cdot {x}^{2}}{z} + \frac{y}{z}}}{x} \]

   5. Simplified 1.0

      \[\leadsto \frac{\color{blue}{\frac{y}{z} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}}{x} \]
      Proof
      (*.f64 (/.f64 y z) (+.f64 1 (*.f64 1/2 (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (+.f64 1 (*.f64 1/2 (Rewrite<= unpow2_binary64 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 y z) 1) (*.f64 (/.f64 y z) (*.f64 1/2 (pow.f64 x 2))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (Rewrite=> *-rgt-identity_binary64 (/.f64 y z)) (*.f64 (/.f64 y z) (*.f64 1/2 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y z) (*.f64 (/.f64 y z) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 x 2) 1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y z) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 y z) (pow.f64 x 2)) 1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y z) (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y (pow.f64 x 2)) z)) 1/2)): 0 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 y z) (Rewrite<= *-commutative_binary64 (*.f64 1/2 (/.f64 (*.f64 y (pow.f64 x 2)) z)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (*.f64 y (pow.f64 x 2)) z)) (/.f64 y z))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{-34}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{x}\\ \mathbf{elif}\;y \leq 10^{-40}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	1480

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{+17}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} + y \cdot \left(\frac{x}{z} \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.0
Cost	1480

\[\begin{array}{l} t_0 := y \cdot \left(\frac{x}{z} \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ t_1 := t_0 + \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{y}{z}}{x} + t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	1.5
Cost	968

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{+17}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.3
Cost	968

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{+17}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{x}\\ \mathbf{elif}\;y \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

Alternative 7
Error	1.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 8
Error	1.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

Alternative 9
Error	1.7
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	1.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 11
Error	8.1
Cost	320

\[\frac{\frac{y}{z}}{x} \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))