Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 8.1 → 0.4

Time: 11.2s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;t_0 \leq 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

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 (* x z))
     (if (<= t_0 1e-47) t_0 (* (cosh x) (/ (/ y z) 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 / (x * z);
	} else if (t_0 <= 1e-47) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * ((y / z) / 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 / (x * z);
	} else if (t_0 <= 1e-47) {
		tmp = t_0;
	} else {
		tmp = Math.cosh(x) * ((y / z) / 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 / (x * z)
	elif t_0 <= 1e-47:
		tmp = t_0
	else:
		tmp = math.cosh(x) * ((y / z) / 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(y / Float64(x * z));
	elseif (t_0 <= 1e-47)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * Float64(Float64(y / z) / 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 / (x * z);
	elseif (t_0 <= 1e-47)
		tmp = t_0;
	else
		tmp = cosh(x) * ((y / z) / 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[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-47], t$95$0, N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	8.1
Target	0.4
Herbie	0.4

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

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 0.5

      \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]
      Proof
      (*.f64 (/.f64 (cosh.f64 x) (*.f64 x z)) y): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cosh.f64 x) (Rewrite<= *-commutative_binary64 (*.f64 z x))) y): 0 points increase in error, 2 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cosh.f64 x) y) (*.f64 z x))): 2 points increase in error, 2 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (cosh.f64 x) z) (/.f64 y x))): 0 points increase in error, 3 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)): 1 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 0.7

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

   if -inf.0 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.9999999999999997e-48

   1. Initial program 0.2

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

   if 9.9999999999999997e-48 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 12.0

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

   2. Simplified 0.6

      \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{z}}{x}} \]
      Proof
      (*.f64 (/.f64 (cosh.f64 x) (*.f64 x z)) y): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cosh.f64 x) (Rewrite<= *-commutative_binary64 (*.f64 z x))) y): 0 points increase in error, 2 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cosh.f64 x) y) (*.f64 z x))): 2 points increase in error, 2 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (cosh.f64 x) z) (/.f64 y x))): 0 points increase in error, 3 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)): 1 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -\infty:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{-47}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-107} \lor \neg \left(y \leq 2.1 \cdot 10^{-49}\right):\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-107}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array} \]

Alternative 3
Error	1.4
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z \leq -5300000 \lor \neg \left(z \leq 7.5 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 4
Error	1.5
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-20} \lor \neg \left(z \leq 6 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \end{array} \]

Alternative 5
Error	1.5
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -200000 \lor \neg \left(z \leq 2.3 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 6
Error	1.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+57} \lor \neg \left(y \leq 5.4 \cdot 10^{-47}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 7
Error	1.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2000000000 \lor \neg \left(z \leq 2.8 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 8
Error	2.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 9
Error	2.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{z}}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 10
Error	8.2
Cost	320

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

Reproduce

herbie shell --seed 2022343 
(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))