Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 0.6

Time: 7.3s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ t_1 := \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+267}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \end{array} \]

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)) (t_1 (/ (/ y z) x)))
   (if (<= t_0 -2e+267)
     t_1
     (if (<= t_0 1e-40) t_0 (+ t_1 (* 0.5 (/ 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 t_1 = (y / z) / x;
	double tmp;
	if (t_0 <= -2e+267) {
		tmp = t_1;
	} else if (t_0 <= 1e-40) {
		tmp = t_0;
	} else {
		tmp = t_1 + (0.5 * (y / (z / x)));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (cosh(x) * (y / x)) / z
    t_1 = (y / z) / x
    if (t_0 <= (-2d+267)) then
        tmp = t_1
    else if (t_0 <= 1d-40) then
        tmp = t_0
    else
        tmp = t_1 + (0.5d0 * (y / (z / x)))
    end if
    code = tmp
end function

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 t_1 = (y / z) / x;
	double tmp;
	if (t_0 <= -2e+267) {
		tmp = t_1;
	} else if (t_0 <= 1e-40) {
		tmp = t_0;
	} else {
		tmp = t_1 + (0.5 * (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
	t_1 = (y / z) / x
	tmp = 0
	if t_0 <= -2e+267:
		tmp = t_1
	elif t_0 <= 1e-40:
		tmp = t_0
	else:
		tmp = t_1 + (0.5 * (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)
	t_1 = Float64(Float64(y / z) / x)
	tmp = 0.0
	if (t_0 <= -2e+267)
		tmp = t_1;
	elseif (t_0 <= 1e-40)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(0.5 * Float64(y / Float64(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;
	t_1 = (y / z) / x;
	tmp = 0.0;
	if (t_0 <= -2e+267)
		tmp = t_1;
	elseif (t_0 <= 1e-40)
		tmp = t_0;
	else
		tmp = t_1 + (0.5 * (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]}, Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+267], t$95$1, If[LessEqual[t$95$0, 1e-40], t$95$0, N[(t$95$1 + N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	7.7
Target	0.4
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -1.9999999999999999e267

   1. Initial program 44.2

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

   2. Simplified 44.3

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}} \]
      Proof
      (/.f64 (cosh.f64 x) (/.f64 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)): 16 points increase in error, 46 points decrease in error

   3. Taylor expanded in x around 0 9.4

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

   4. Simplified 0.6

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]
      Proof
      (/.f64 (/.f64 y z) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 y (*.f64 z x))): 76 points increase in error, 62 points decrease in error

   if -1.9999999999999999e267 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.9999999999999993e-41

   1. Initial program 0.2

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

   if 9.9999999999999993e-41 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 11.6

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

   2. Taylor expanded in x around 0 11.0

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

   3. Simplified 1.3

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x} + 0.5 \cdot \frac{y}{\frac{z}{x}}} \]
      Proof
      (+.f64 (/.f64 (/.f64 y z) x) (*.f64 1/2 (/.f64 y (/.f64 z x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/r*_binary64 (/.f64 y (*.f64 z x))) (*.f64 1/2 (/.f64 y (/.f64 z x)))): 77 points increase in error, 60 points decrease in error
      (+.f64 (/.f64 y (*.f64 z x)) (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) z)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -2 \cdot 10^{+267}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{-40}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	7112

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.1
Cost	1096

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.5
Cost	968

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.3
Cost	968

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{if}\;y \leq -0.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.7
Cost	584

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.5
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -9.8 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x}}{z}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 8
Error	8.3
Cost	320

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

Reproduce

herbie shell --seed 2022325 
(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))