exp2 (problem 3.3.7)

Average Accuracy: 26.2% → 50.5%

Time: 8.2s

Precision: binary64

Cost: 192

Math

\[\left(e^{x} - 2\right) + e^{-x} \]

↓

\[x \cdot x \]

FPCore

(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))

↓

(FPCore (x) :precision binary64 (* x x))

C

double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}

↓

double code(double x) {
	return x * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - 2.0d0) + exp(-x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * x
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 2.0) + Math.exp(-x);
}

↓

public static double code(double x) {
	return x * x;
}

Python

def code(x):
	return (math.exp(x) - 2.0) + math.exp(-x)

↓

def code(x):
	return x * x

Julia

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end

↓

function code(x)
	return Float64(x * x)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 2.0) + exp(-x);
end

↓

function tmp = code(x)
	tmp = x * x;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x * x), $MachinePrecision]

Target

Original	26.2%
Target	49.8%
Herbie	50.5%

\[4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2} \]

Derivation

1. Initial program 29.9%

   \[\left(e^{x} - 2\right) + e^{-x} \]

2. Simplified 30.0%

   \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

3. Taylor expanded in x around 0 55.9%

   \[\leadsto \color{blue}{{x}^{2}} \]

4. Simplified 55.9%

   \[\leadsto \color{blue}{x \cdot x} \]

5. Final simplification 55.9%

   \[\leadsto x \cdot x \]

Alternatives

Alternative 1
Accuracy	25.9%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))