?

Average Error: 2.5 → 0.6
Time: 10.5s
Precision: binary64
Cost: 20425

?

\[\frac{x \cdot \frac{\sin y}{y}}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-171} \lor \neg \left(t_1 \leq 10^{-62}\right):\\ \;\;\;\;\frac{t_1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* x t_0)))
   (if (or (<= t_1 -1e-171) (not (<= t_1 1e-62))) (/ t_1 z) (/ x (/ z t_0)))))
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 = sin(y) / y;
	double t_1 = x * t_0;
	double tmp;
	if ((t_1 <= -1e-171) || !(t_1 <= 1e-62)) {
		tmp = t_1 / z;
	} else {
		tmp = x / (z / t_0);
	}
	return tmp;
}

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 = sin(y) / y
    t_1 = x * t_0
    if ((t_1 <= (-1d-171)) .or. (.not. (t_1 <= 1d-62))) then
        tmp = t_1 / z
    else
        tmp = x / (z / t_0)
    end if
    code = tmp
end function

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 = Math.sin(y) / y;
	double t_1 = x * t_0;
	double tmp;
	if ((t_1 <= -1e-171) || !(t_1 <= 1e-62)) {
		tmp = t_1 / z;
	} else {
		tmp = x / (z / t_0);
	}
	return tmp;
}

def code(x, y, z):
	return (x * (math.sin(y) / y)) / z
↓
def code(x, y, z):
	t_0 = math.sin(y) / y
	t_1 = x * t_0
	tmp = 0
	if (t_1 <= -1e-171) or not (t_1 <= 1e-62):
		tmp = t_1 / z
	else:
		tmp = x / (z / t_0)
	return tmp

function code(x, y, z)
	return Float64(Float64(x * Float64(sin(y) / y)) / z)
end
↓
function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if ((t_1 <= -1e-171) || !(t_1 <= 1e-62))
		tmp = Float64(t_1 / z);
	else
		tmp = Float64(x / Float64(z / t_0));
	end
	return tmp
end

function tmp = code(x, y, z)
	tmp = (x * (sin(y) / y)) / z;
end
↓
function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	t_1 = x * t_0;
	tmp = 0.0;
	if ((t_1 <= -1e-171) || ~((t_1 <= 1e-62)))
		tmp = t_1 / z;
	else
		tmp = x / (z / t_0);
	end
	tmp_2 = tmp;
end

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[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-171], N[Not[LessEqual[t$95$1, 1e-62]], $MachinePrecision]], N[(t$95$1 / z), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
t_0 := \frac{\sin y}{y}\\
t_1 := x \cdot t_0\\
\mathbf{if}\;t_1 \leq -1 \cdot 10^{-171} \lor \neg \left(t_1 \leq 10^{-62}\right):\\
\;\;\;\;\frac{t_1}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{z}{t_0}}\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.5
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation?

  1. Split input into 2 regimes

  2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -9.9999999999999998e-172 or 1e-62 < (*.f64 x (/.f64 (sin.f64 y) y))

    1. Initial program 0.2

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

    if -9.9999999999999998e-172 < (*.f64 x (/.f64 (sin.f64 y) y)) < 1e-62

    1. Initial program 5.1

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

    2. Simplified1.1

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

      [Start]5.1

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

      associate-/l* [=>]1.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-171} \lor \neg \left(x \cdot \frac{\sin y}{y} \leq 10^{-62}\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternatives

Alternative 1
Error2.8
Cost7113
\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-8} \lor \neg \left(y \leq 3.1 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
Alternative 2
Error2.8
Cost6848
\[\frac{x}{z \cdot \frac{y}{\sin y}} \]
Alternative 3
Error2.8
Cost6848
\[\frac{x}{\frac{z}{\frac{\sin y}{y}}} \]
Alternative 4
Error22.6
Cost841
\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
Alternative 5
Error22.6
Cost841
\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;\frac{\frac{x}{\frac{y \cdot y}{6}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
Alternative 6
Error22.6
Cost840
\[\begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(y \cdot z\right)}\\ \end{array} \]
Alternative 7
Error22.6
Cost840
\[\begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y \cdot z}}{y}\\ \end{array} \]
Alternative 8
Error22.6
Cost840
\[\begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;\frac{\frac{x}{\frac{y \cdot y}{6}}}{z}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{y}{6}}}{y}}{z}\\ \end{array} \]
Alternative 9
Error22.9
Cost713
\[\begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+34} \lor \neg \left(y \leq 3.8 \cdot 10^{+55}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
Alternative 10
Error22.4
Cost704
\[\frac{x}{z \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)} \]
Alternative 11
Error28.1
Cost192
\[\frac{x}{z} \]

Error

Reproduce?

herbie shell --seed 2023066 
(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))