Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.6 → 0.3

Time: 9.8s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ t_1 := \frac{t_0}{z}\\ \mathbf{if}\;t_0 \leq -9 \cdot 10^{-273}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ (sin y) y))) (t_1 (/ t_0 z)))
   (if (<= t_0 -9e-273) t_1 (if (<= t_0 0.0) (* x (/ (/ (sin y) z) y)) t_1))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = x * (sin(y) / y);
	double t_1 = t_0 / z;
	double tmp;
	if (t_0 <= -9e-273) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = x * ((sin(y) / z) / y);
	} else {
		tmp = t_1;
	}
	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 = (x * (sin(y) / y)) / 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 = x * (sin(y) / y)
    t_1 = t_0 / z
    if (t_0 <= (-9d-273)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = x * ((sin(y) / z) / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = x * (Math.sin(y) / y);
	double t_1 = t_0 / z;
	double tmp;
	if (t_0 <= -9e-273) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = x * ((Math.sin(y) / z) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = x * (math.sin(y) / y)
	t_1 = t_0 / z
	tmp = 0
	if t_0 <= -9e-273:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = x * ((math.sin(y) / z) / y)
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(x * Float64(sin(y) / y))
	t_1 = Float64(t_0 / z)
	tmp = 0.0
	if (t_0 <= -9e-273)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(Float64(sin(y) / z) / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = x * (sin(y) / y);
	t_1 = t_0 / z;
	tmp = 0.0;
	if (t_0 <= -9e-273)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = x * ((sin(y) / z) / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / z), $MachinePrecision]}, If[LessEqual[t$95$0, -9e-273], t$95$1, If[LessEqual[t$95$0, 0.0], N[(x * N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	2.6
Target	0.2
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -8.99999999999999921e-273 or -0.0 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 0.3

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

   if -8.99999999999999921e-273 < (*.f64 x (/.f64 (sin.f64 y) y)) < -0.0

   1. Initial program 15.2

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

   2. Simplified 15.9

      \[\leadsto \color{blue}{\frac{\sin y}{z \cdot \frac{y}{x}}} \]
      Proof
      (/.f64 (sin.f64 y) (*.f64 z (/.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (sin.f64 y) (/.f64 y x)) z)): 15 points increase in error, 47 points decrease in error
      (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (sin.f64 y) y) x)) z): 12 points increase in error, 48 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x (/.f64 (sin.f64 y) y))) z): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{z}}{y} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -9 \cdot 10^{-273}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \leq 0:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.8
Cost	7112

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{if}\;y \leq -7.453258451605268 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.157821438955228 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -7.453258451605268 \cdot 10^{-6}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{elif}\;y \leq 4.157821438955228 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \end{array} \]

Alternative 3
Error	2.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -7.453258451605268 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \sin y}{y \cdot z}\\ \mathbf{elif}\;y \leq 4.157821438955228 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \end{array} \]

Alternative 4
Error	2.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -7.453258451605268 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \sin y}{y \cdot z}\\ \mathbf{elif}\;y \leq 4.157821438955228 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 5
Error	28.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq 3.3360514296809222 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 3.61271154004458 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 6
Error	27.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -3.729805295106613 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 3.3360514296809222 \cdot 10^{-93}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y \cdot z}\\ \end{array} \]

Alternative 7
Error	23.3
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{x}{y \cdot z}\\ \mathbf{if}\;y \leq -4.896529830071695 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.172391021827818 \cdot 10^{+74}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	22.9
Cost	712

\[\begin{array}{l} t_0 := \frac{y}{z \cdot \frac{y}{x}}\\ \mathbf{if}\;y \leq -4.896529830071695 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3872364812726551 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	22.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4.896529830071695 \cdot 10^{+69}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 173683702205.44278:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \end{array} \]

Alternative 10
Error	28.0
Cost	192

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

Reproduce

herbie shell --seed 2022308 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))